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Properties

Label 2-63-63.23-c2-0-0
Degree 22
Conductor 6363
Sign −0.915+0.401i-0.915 + 0.401i
Analytic cond. 1.716621.71662
Root an. cond. 1.310201.31020
Motivic weight 22
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.22i·2-s + (−2.32 − 1.89i)3-s − 6.39·4-s + (−4.79 + 2.76i)5-s + (6.10 − 7.50i)6-s + (−6.99 − 0.206i)7-s − 7.70i·8-s + (1.82 + 8.81i)9-s + (−8.91 − 15.4i)10-s + (15.3 + 8.84i)11-s + (14.8 + 12.1i)12-s + (2.03 − 3.52i)13-s + (0.665 − 22.5i)14-s + (16.3 + 2.63i)15-s − 0.715·16-s + (−14.3 + 8.27i)17-s + ⋯
L(s)  = 1  + 1.61i·2-s + (−0.775 − 0.631i)3-s − 1.59·4-s + (−0.958 + 0.553i)5-s + (1.01 − 1.25i)6-s + (−0.999 − 0.0294i)7-s − 0.963i·8-s + (0.203 + 0.979i)9-s + (−0.891 − 1.54i)10-s + (1.39 + 0.804i)11-s + (1.23 + 1.00i)12-s + (0.156 − 0.271i)13-s + (0.0475 − 1.61i)14-s + (1.09 + 0.175i)15-s − 0.0447·16-s + (−0.843 + 0.486i)17-s + ⋯

Functional equation

Λ(s)=(63s/2 ΓC(s) L(s)=((−0.915+0.401i) Λ‾(3−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(3-s) \end{aligned}
Λ(s)=(63s/2 ΓC(s+1) L(s)=((−0.915+0.401i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 6363    =    32⋅73^{2} \cdot 7
Sign: −0.915+0.401i-0.915 + 0.401i
Analytic conductor: 1.716621.71662
Root analytic conductor: 1.310201.31020
Motivic weight: 22
Rational: no
Arithmetic: yes
Character: χ63(23,⋅)\chi_{63} (23, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 63, ( :1), −0.915+0.401i)(2,\ 63,\ (\ :1),\ -0.915 + 0.401i)

Particular Values

L(32)L(\frac{3}{2}) ≈\approx 0.0982891−0.469184i0.0982891 - 0.469184i
L(12)L(\frac12) ≈\approx 0.0982891−0.469184i0.0982891 - 0.469184i
L(2)L(2) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad3 1+(2.32+1.89i)T 1 + (2.32 + 1.89i)T
7 1+(6.99+0.206i)T 1 + (6.99 + 0.206i)T
good2 1−3.22iT−4T2 1 - 3.22iT - 4T^{2}
5 1+(4.79−2.76i)T+(12.5−21.6i)T2 1 + (4.79 - 2.76i)T + (12.5 - 21.6i)T^{2}
11 1+(−15.3−8.84i)T+(60.5+104.i)T2 1 + (-15.3 - 8.84i)T + (60.5 + 104. i)T^{2}
13 1+(−2.03+3.52i)T+(−84.5−146.i)T2 1 + (-2.03 + 3.52i)T + (-84.5 - 146. i)T^{2}
17 1+(14.3−8.27i)T+(144.5−250.i)T2 1 + (14.3 - 8.27i)T + (144.5 - 250. i)T^{2}
19 1+(3.92−6.79i)T+(−180.5−312.i)T2 1 + (3.92 - 6.79i)T + (-180.5 - 312. i)T^{2}
23 1+(−8.71+5.03i)T+(264.5−458.i)T2 1 + (-8.71 + 5.03i)T + (264.5 - 458. i)T^{2}
29 1+(39.9−23.0i)T+(420.5−728.i)T2 1 + (39.9 - 23.0i)T + (420.5 - 728. i)T^{2}
31 1−29.6T+961T2 1 - 29.6T + 961T^{2}
37 1+(−15.5+27.0i)T+(−684.5−1.18e3i)T2 1 + (-15.5 + 27.0i)T + (-684.5 - 1.18e3i)T^{2}
41 1+(−27.8−16.0i)T+(840.5+1.45e3i)T2 1 + (-27.8 - 16.0i)T + (840.5 + 1.45e3i)T^{2}
43 1+(−3.35−5.80i)T+(−924.5+1.60e3i)T2 1 + (-3.35 - 5.80i)T + (-924.5 + 1.60e3i)T^{2}
47 1−16.4iT−2.20e3T2 1 - 16.4iT - 2.20e3T^{2}
53 1+(32.5−18.8i)T+(1.40e3−2.43e3i)T2 1 + (32.5 - 18.8i)T + (1.40e3 - 2.43e3i)T^{2}
59 1−95.0iT−3.48e3T2 1 - 95.0iT - 3.48e3T^{2}
61 1+73.7T+3.72e3T2 1 + 73.7T + 3.72e3T^{2}
67 1+12.1T+4.48e3T2 1 + 12.1T + 4.48e3T^{2}
71 1−20.0iT−5.04e3T2 1 - 20.0iT - 5.04e3T^{2}
73 1+(11.4+19.9i)T+(−2.66e3+4.61e3i)T2 1 + (11.4 + 19.9i)T + (-2.66e3 + 4.61e3i)T^{2}
79 1−138.T+6.24e3T2 1 - 138.T + 6.24e3T^{2}
83 1+(−13.6+7.90i)T+(3.44e3−5.96e3i)T2 1 + (-13.6 + 7.90i)T + (3.44e3 - 5.96e3i)T^{2}
89 1+(46.9+27.1i)T+(3.96e3+6.85e3i)T2 1 + (46.9 + 27.1i)T + (3.96e3 + 6.85e3i)T^{2}
97 1+(−86.1−149.i)T+(−4.70e3+8.14e3i)T2 1 + (-86.1 - 149. i)T + (-4.70e3 + 8.14e3i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−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

−15.42545571160667637399947059231, −14.64084946730940203716364168577, −13.27260012910896755059679621080, −12.21544173031453596940561468827, −10.94514640011731605373566170680, −9.177291351558186116172600023674, −7.65306353522304996048005953645, −6.84198526327876333673349553119, −6.08671492236474833199556270739, −4.23145641032519137339379246079, 0.49399152011033700893973996607, 3.52656214496322384633372660971, 4.41644864369809905289471243342, 6.46593421671320551856048375501, 8.944481569715082385164913646858, 9.617743818522722925979577161089, 11.09873643314263035563508126734, 11.62539484095168217076057378175, 12.47301022923135225109538062794, 13.59425372436257652903050226272

Graph of the ZZ-function along the critical line