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Properties

Label 2-308-77.54-c1-0-4
Degree 22
Conductor 308308
Sign 0.685+0.728i0.685 + 0.728i
Analytic cond. 2.459392.45939
Root an. cond. 1.568241.56824
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.14 + 1.23i)3-s + (−0.349 − 0.201i)5-s + (−0.938 − 2.47i)7-s + (1.56 − 2.71i)9-s + (0.0145 − 3.31i)11-s + 3.56·13-s + 0.999·15-s + (3.51 + 6.09i)17-s + (3.51 − 6.09i)19-s + (5.07 + 4.14i)21-s + (1.29 − 2.24i)23-s + (−2.41 − 4.18i)25-s + 0.341i·27-s − 7.90i·29-s + (−6.85 + 3.95i)31-s + ⋯
L(s)  = 1  + (−1.23 + 0.715i)3-s + (−0.156 − 0.0902i)5-s + (−0.354 − 0.934i)7-s + (0.522 − 0.905i)9-s + (0.00437 − 0.999i)11-s + 0.989·13-s + 0.258·15-s + (0.852 + 1.47i)17-s + (0.806 − 1.39i)19-s + (1.10 + 0.904i)21-s + (0.270 − 0.468i)23-s + (−0.483 − 0.837i)25-s + 0.0657i·27-s − 1.46i·29-s + (−1.23 + 0.710i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((0.685+0.728i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((0.685+0.728i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.685 + 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 0.685+0.728i0.685 + 0.728i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(285,⋅)\chi_{308} (285, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), 0.685+0.728i)(2,\ 308,\ (\ :1/2),\ 0.685 + 0.728i)

Particular Values

L(1)L(1) ≈\approx 0.682210−0.294735i0.682210 - 0.294735i
L(12)L(\frac12) ≈\approx 0.682210−0.294735i0.682210 - 0.294735i
L(32)L(\frac{3}{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)
bad2 1 1
7 1+(0.938+2.47i)T 1 + (0.938 + 2.47i)T
11 1+(−0.0145+3.31i)T 1 + (-0.0145 + 3.31i)T
good3 1+(2.14−1.23i)T+(1.5−2.59i)T2 1 + (2.14 - 1.23i)T + (1.5 - 2.59i)T^{2}
5 1+(0.349+0.201i)T+(2.5+4.33i)T2 1 + (0.349 + 0.201i)T + (2.5 + 4.33i)T^{2}
13 1−3.56T+13T2 1 - 3.56T + 13T^{2}
17 1+(−3.51−6.09i)T+(−8.5+14.7i)T2 1 + (-3.51 - 6.09i)T + (-8.5 + 14.7i)T^{2}
19 1+(−3.51+6.09i)T+(−9.5−16.4i)T2 1 + (-3.51 + 6.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(−1.29+2.24i)T+(−11.5−19.9i)T2 1 + (-1.29 + 2.24i)T + (-11.5 - 19.9i)T^{2}
29 1+7.90iT−29T2 1 + 7.90iT - 29T^{2}
31 1+(6.85−3.95i)T+(15.5−26.8i)T2 1 + (6.85 - 3.95i)T + (15.5 - 26.8i)T^{2}
37 1+(−2.86+4.96i)T+(−18.5−32.0i)T2 1 + (-2.86 + 4.96i)T + (-18.5 - 32.0i)T^{2}
41 1+3.56T+41T2 1 + 3.56T + 41T^{2}
43 1+0.937iT−43T2 1 + 0.937iT - 43T^{2}
47 1+(−7.49−4.32i)T+(23.5+40.7i)T2 1 + (-7.49 - 4.32i)T + (23.5 + 40.7i)T^{2}
53 1+(−0.992−1.71i)T+(−26.5+45.8i)T2 1 + (-0.992 - 1.71i)T + (-26.5 + 45.8i)T^{2}
59 1+(7.94−4.58i)T+(29.5−51.0i)T2 1 + (7.94 - 4.58i)T + (29.5 - 51.0i)T^{2}
61 1+(−2.89+5.01i)T+(−30.5−52.8i)T2 1 + (-2.89 + 5.01i)T + (-30.5 - 52.8i)T^{2}
67 1+(−0.426−0.738i)T+(−33.5+58.0i)T2 1 + (-0.426 - 0.738i)T + (-33.5 + 58.0i)T^{2}
71 1−0.990T+71T2 1 - 0.990T + 71T^{2}
73 1+(6.89+11.9i)T+(−36.5+63.2i)T2 1 + (6.89 + 11.9i)T + (-36.5 + 63.2i)T^{2}
79 1+(−3.95−2.28i)T+(39.5+68.4i)T2 1 + (-3.95 - 2.28i)T + (39.5 + 68.4i)T^{2}
83 1+1.24T+83T2 1 + 1.24T + 83T^{2}
89 1+(1.83+1.06i)T+(44.5+77.0i)T2 1 + (1.83 + 1.06i)T + (44.5 + 77.0i)T^{2}
97 1+7.57iT−97T2 1 + 7.57iT - 97T^{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

−11.23968521391881920964195549560, −10.80756110893048303962639109812, −10.04999349537897453458304009671, −8.862950053743659760500489827360, −7.67916259165491573516169063488, −6.32578146591673626878301693111, −5.72160724754435150796505081777, −4.40151554996092555135154099917, −3.50999733470198625959197051825, −0.69884957718535691083527582901, 1.50064323044594585941767350321, 3.38499876792192849247414129821, 5.26203981021514092144060654182, 5.74890215635876039417944769848, 6.93849410513162325678088504582, 7.65664097908544381505141576127, 9.125821078390038877345488589009, 10.00041828970656040907559088245, 11.26321551690612734549896680352, 11.85231519961283838674462579074

Graph of the ZZ-function along the critical line