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Properties

Label 2-308-44.43-c1-0-18
Degree 22
Conductor 308308
Sign 0.954−0.297i0.954 - 0.297i
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  + (0.963 − 1.03i)2-s + 3.04i·3-s + (−0.145 − 1.99i)4-s + 2.80·5-s + (3.15 + 2.93i)6-s + 7-s + (−2.20 − 1.77i)8-s − 6.29·9-s + (2.70 − 2.90i)10-s + (0.755 + 3.22i)11-s + (6.08 − 0.442i)12-s − 5.37i·13-s + (0.963 − 1.03i)14-s + 8.55i·15-s + (−3.95 + 0.578i)16-s + 5.95i·17-s + ⋯
L(s)  = 1  + (0.680 − 0.732i)2-s + 1.76i·3-s + (−0.0725 − 0.997i)4-s + 1.25·5-s + (1.28 + 1.19i)6-s + 0.377·7-s + (−0.779 − 0.626i)8-s − 2.09·9-s + (0.854 − 0.918i)10-s + (0.227 + 0.973i)11-s + (1.75 − 0.127i)12-s − 1.49i·13-s + (0.257 − 0.276i)14-s + 2.20i·15-s + (−0.989 + 0.144i)16-s + 1.44i·17-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((0.954−0.297i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((0.954−0.297i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.297i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: 0.954−0.297i0.954 - 0.297i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(43,⋅)\chi_{308} (43, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), 0.954−0.297i)(2,\ 308,\ (\ :1/2),\ 0.954 - 0.297i)

Particular Values

L(1)L(1) ≈\approx 2.08417+0.317665i2.08417 + 0.317665i
L(12)L(\frac12) ≈\approx 2.08417+0.317665i2.08417 + 0.317665i
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+(−0.963+1.03i)T 1 + (-0.963 + 1.03i)T
7 1−T 1 - T
11 1+(−0.755−3.22i)T 1 + (-0.755 - 3.22i)T
good3 1−3.04iT−3T2 1 - 3.04iT - 3T^{2}
5 1−2.80T+5T2 1 - 2.80T + 5T^{2}
13 1+5.37iT−13T2 1 + 5.37iT - 13T^{2}
17 1−5.95iT−17T2 1 - 5.95iT - 17T^{2}
19 1+0.325T+19T2 1 + 0.325T + 19T^{2}
23 1+1.56iT−23T2 1 + 1.56iT - 23T^{2}
29 1+4.39iT−29T2 1 + 4.39iT - 29T^{2}
31 1+3.00iT−31T2 1 + 3.00iT - 31T^{2}
37 1−4.11T+37T2 1 - 4.11T + 37T^{2}
41 1+1.41iT−41T2 1 + 1.41iT - 41T^{2}
43 1+12.4T+43T2 1 + 12.4T + 43T^{2}
47 1+9.32iT−47T2 1 + 9.32iT - 47T^{2}
53 1+4.91T+53T2 1 + 4.91T + 53T^{2}
59 1−4.96iT−59T2 1 - 4.96iT - 59T^{2}
61 1+10.7iT−61T2 1 + 10.7iT - 61T^{2}
67 1−2.71iT−67T2 1 - 2.71iT - 67T^{2}
71 1+1.79iT−71T2 1 + 1.79iT - 71T^{2}
73 1−7.05iT−73T2 1 - 7.05iT - 73T^{2}
79 1+1.63T+79T2 1 + 1.63T + 79T^{2}
83 1+5.50T+83T2 1 + 5.50T + 83T^{2}
89 1−5.20T+89T2 1 - 5.20T + 89T^{2}
97 1−16.1T+97T2 1 - 16.1T + 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

−11.51585837304928823035451211085, −10.46008384927860398426874927852, −10.15591256628819277011359826699, −9.515558828995429433634891825030, −8.374183170864236601430425261747, −6.17890545247397969438388876168, −5.42431734298807919593719511037, −4.58523986663970937311307261881, −3.51599209506219783550498593221, −2.14619786898709527752320022483, 1.68147599820180683789020796450, 2.92427075267274348434556753758, 4.98711659462077989702910159210, 6.04131975082701357774111931637, 6.64214564959893221475654461748, 7.44897325795429686652267710160, 8.577007646200970247258632794949, 9.329250777860533703563289912222, 11.33841197973281248633552464724, 11.82351702081016615825272758926

Graph of the ZZ-function along the critical line