Properties

Label 2-22-11.5-c13-0-1
Degree 22
Conductor 2222
Sign 0.757+0.652i-0.757 + 0.652i
Analytic cond. 23.590823.5908
Root an. cond. 4.857034.85703
Motivic weight 1313
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (19.7 + 60.8i)2-s + (1.19e3 + 866. i)3-s + (−3.31e3 + 2.40e3i)4-s + (−1.62e4 + 4.99e4i)5-s + (−2.91e4 + 8.97e4i)6-s + (−2.03e5 + 1.47e5i)7-s + (−2.12e5 − 1.54e5i)8-s + (1.79e5 + 5.52e5i)9-s − 3.36e6·10-s + (−7.55e5 − 5.82e6i)11-s − 6.04e6·12-s + (2.64e6 + 8.13e6i)13-s + (−1.29e7 − 9.44e6i)14-s + (−6.26e7 + 4.55e7i)15-s + (5.18e6 − 1.59e7i)16-s + (1.00e7 − 3.09e7i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (0.944 + 0.686i)3-s + (−0.404 + 0.293i)4-s + (−0.464 + 1.42i)5-s + (−0.255 + 0.785i)6-s + (−0.652 + 0.474i)7-s + (−0.286 − 0.207i)8-s + (0.112 + 0.346i)9-s − 1.06·10-s + (−0.128 − 0.991i)11-s − 0.584·12-s + (0.151 + 0.467i)13-s + (−0.461 − 0.335i)14-s + (−1.42 + 1.03i)15-s + (0.0772 − 0.237i)16-s + (0.101 − 0.311i)17-s + ⋯

Functional equation

Λ(s)=(22s/2ΓC(s)L(s)=((0.757+0.652i)Λ(14s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(14-s) \end{aligned}
Λ(s)=(22s/2ΓC(s+13/2)L(s)=((0.757+0.652i)Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 2222    =    2112 \cdot 11
Sign: 0.757+0.652i-0.757 + 0.652i
Analytic conductor: 23.590823.5908
Root analytic conductor: 4.857034.85703
Motivic weight: 1313
Rational: no
Arithmetic: yes
Character: χ22(5,)\chi_{22} (5, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 22, ( :13/2), 0.757+0.652i)(2,\ 22,\ (\ :13/2),\ -0.757 + 0.652i)

Particular Values

L(7)L(7) \approx 1.6549649081.654964908
L(12)L(\frac12) \approx 1.6549649081.654964908
L(152)L(\frac{15}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(19.760.8i)T 1 + (-19.7 - 60.8i)T
11 1+(7.55e5+5.82e6i)T 1 + (7.55e5 + 5.82e6i)T
good3 1+(1.19e3866.i)T+(4.92e5+1.51e6i)T2 1 + (-1.19e3 - 866. i)T + (4.92e5 + 1.51e6i)T^{2}
5 1+(1.62e44.99e4i)T+(9.87e87.17e8i)T2 1 + (1.62e4 - 4.99e4i)T + (-9.87e8 - 7.17e8i)T^{2}
7 1+(2.03e51.47e5i)T+(2.99e109.21e10i)T2 1 + (2.03e5 - 1.47e5i)T + (2.99e10 - 9.21e10i)T^{2}
13 1+(2.64e68.13e6i)T+(2.45e14+1.78e14i)T2 1 + (-2.64e6 - 8.13e6i)T + (-2.45e14 + 1.78e14i)T^{2}
17 1+(1.00e7+3.09e7i)T+(8.01e155.82e15i)T2 1 + (-1.00e7 + 3.09e7i)T + (-8.01e15 - 5.82e15i)T^{2}
19 1+(2.79e82.02e8i)T+(1.29e16+3.99e16i)T2 1 + (-2.79e8 - 2.02e8i)T + (1.29e16 + 3.99e16i)T^{2}
23 1+2.93e8T+5.04e17T2 1 + 2.93e8T + 5.04e17T^{2}
29 1+(2.88e92.09e9i)T+(3.17e189.75e18i)T2 1 + (2.88e9 - 2.09e9i)T + (3.17e18 - 9.75e18i)T^{2}
31 1+(3.26e8+1.00e9i)T+(1.97e19+1.43e19i)T2 1 + (3.26e8 + 1.00e9i)T + (-1.97e19 + 1.43e19i)T^{2}
37 1+(4.20e93.05e9i)T+(7.52e192.31e20i)T2 1 + (4.20e9 - 3.05e9i)T + (7.52e19 - 2.31e20i)T^{2}
41 1+(2.43e10+1.77e10i)T+(2.85e20+8.79e20i)T2 1 + (2.43e10 + 1.77e10i)T + (2.85e20 + 8.79e20i)T^{2}
43 1+6.71e10T+1.71e21T2 1 + 6.71e10T + 1.71e21T^{2}
47 1+(7.64e105.55e10i)T+(1.68e21+5.19e21i)T2 1 + (-7.64e10 - 5.55e10i)T + (1.68e21 + 5.19e21i)T^{2}
53 1+(6.78e102.08e11i)T+(2.10e22+1.53e22i)T2 1 + (-6.78e10 - 2.08e11i)T + (-2.10e22 + 1.53e22i)T^{2}
59 1+(3.13e11+2.28e11i)T+(3.24e229.98e22i)T2 1 + (-3.13e11 + 2.28e11i)T + (3.24e22 - 9.98e22i)T^{2}
61 1+(1.36e114.19e11i)T+(1.30e239.51e22i)T2 1 + (1.36e11 - 4.19e11i)T + (-1.30e23 - 9.51e22i)T^{2}
67 1+1.03e12T+5.48e23T2 1 + 1.03e12T + 5.48e23T^{2}
71 1+(6.07e111.86e12i)T+(9.42e236.84e23i)T2 1 + (6.07e11 - 1.86e12i)T + (-9.42e23 - 6.84e23i)T^{2}
73 1+(3.11e112.26e11i)T+(5.16e231.59e24i)T2 1 + (3.11e11 - 2.26e11i)T + (5.16e23 - 1.59e24i)T^{2}
79 1+(3.04e11+9.35e11i)T+(3.77e24+2.74e24i)T2 1 + (3.04e11 + 9.35e11i)T + (-3.77e24 + 2.74e24i)T^{2}
83 1+(4.10e11+1.26e12i)T+(7.17e245.21e24i)T2 1 + (-4.10e11 + 1.26e12i)T + (-7.17e24 - 5.21e24i)T^{2}
89 11.62e12T+2.19e25T2 1 - 1.62e12T + 2.19e25T^{2}
97 1+(3.05e129.39e12i)T+(5.44e25+3.95e25i)T2 1 + (-3.05e12 - 9.39e12i)T + (-5.44e25 + 3.95e25i)T^{2}
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   L(s)=p j=12(1αj,pps)1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−15.52097837415761155434518093541, −14.53384187749266485407948645087, −13.77722816188209064843406591509, −11.74208247691684275832909140543, −10.10661213084280514386338267608, −8.831783889549821927702768901795, −7.40379213668336355860704921039, −5.95726197856096274725124118757, −3.64503569489932124833072782418, −3.00290695997858487919666422350, 0.46059581780771040970308573645, 1.77699264427219907689491702731, 3.45016100409143803961530645710, 4.98877106454144206780554108531, 7.41904104274137910275167980343, 8.656164356417693667584251109739, 9.863740609190886174892613424734, 11.90305197104951954541509812397, 13.01258840736713622298333070223, 13.51484468698647899040264426243

Graph of the ZZ-function along the critical line