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Properties

Label 2-308-11.3-c1-0-0
Degree 22
Conductor 308308
Sign −0.642−0.766i-0.642 - 0.766i
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  + (1 + 3.07i)3-s + (−1.61 − 1.17i)5-s + (−0.309 + 0.951i)7-s + (−6.04 + 4.39i)9-s + (2.19 + 2.48i)11-s + (−2.61 + 1.90i)13-s + (2 − 6.15i)15-s + (3.23 + 2.35i)17-s + (−0.618 − 1.90i)19-s − 3.23·21-s + 0.618·23-s + (−0.309 − 0.951i)25-s + (−11.7 − 8.50i)27-s + (−0.263 + 0.812i)29-s + (3.61 − 2.62i)31-s + ⋯
L(s)  = 1  + (0.577 + 1.77i)3-s + (−0.723 − 0.525i)5-s + (−0.116 + 0.359i)7-s + (−2.01 + 1.46i)9-s + (0.660 + 0.750i)11-s + (−0.726 + 0.527i)13-s + (0.516 − 1.58i)15-s + (0.784 + 0.570i)17-s + (−0.141 − 0.436i)19-s − 0.706·21-s + 0.128·23-s + (−0.0618 − 0.190i)25-s + (−2.25 − 1.63i)27-s + (−0.0490 + 0.150i)29-s + (0.649 − 0.472i)31-s + ⋯

Functional equation

Λ(s)=(308s/2 ΓC(s) L(s)=((−0.642−0.766i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(308s/2 ΓC(s+1/2) L(s)=((−0.642−0.766i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 308308    =    22⋅7⋅112^{2} \cdot 7 \cdot 11
Sign: −0.642−0.766i-0.642 - 0.766i
Analytic conductor: 2.459392.45939
Root analytic conductor: 1.568241.56824
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ308(113,⋅)\chi_{308} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 308, ( :1/2), −0.642−0.766i)(2,\ 308,\ (\ :1/2),\ -0.642 - 0.766i)

Particular Values

L(1)L(1) ≈\approx 0.526406+1.12847i0.526406 + 1.12847i
L(12)L(\frac12) ≈\approx 0.526406+1.12847i0.526406 + 1.12847i
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+(0.309−0.951i)T 1 + (0.309 - 0.951i)T
11 1+(−2.19−2.48i)T 1 + (-2.19 - 2.48i)T
good3 1+(−1−3.07i)T+(−2.42+1.76i)T2 1 + (-1 - 3.07i)T + (-2.42 + 1.76i)T^{2}
5 1+(1.61+1.17i)T+(1.54+4.75i)T2 1 + (1.61 + 1.17i)T + (1.54 + 4.75i)T^{2}
13 1+(2.61−1.90i)T+(4.01−12.3i)T2 1 + (2.61 - 1.90i)T + (4.01 - 12.3i)T^{2}
17 1+(−3.23−2.35i)T+(5.25+16.1i)T2 1 + (-3.23 - 2.35i)T + (5.25 + 16.1i)T^{2}
19 1+(0.618+1.90i)T+(−15.3+11.1i)T2 1 + (0.618 + 1.90i)T + (-15.3 + 11.1i)T^{2}
23 1−0.618T+23T2 1 - 0.618T + 23T^{2}
29 1+(0.263−0.812i)T+(−23.4−17.0i)T2 1 + (0.263 - 0.812i)T + (-23.4 - 17.0i)T^{2}
31 1+(−3.61+2.62i)T+(9.57−29.4i)T2 1 + (-3.61 + 2.62i)T + (9.57 - 29.4i)T^{2}
37 1+(−3.04+9.37i)T+(−29.9−21.7i)T2 1 + (-3.04 + 9.37i)T + (-29.9 - 21.7i)T^{2}
41 1+(−3.61−11.1i)T+(−33.1+24.0i)T2 1 + (-3.61 - 11.1i)T + (-33.1 + 24.0i)T^{2}
43 1−10.8T+43T2 1 - 10.8T + 43T^{2}
47 1+(−3.09−9.51i)T+(−38.0+27.6i)T2 1 + (-3.09 - 9.51i)T + (-38.0 + 27.6i)T^{2}
53 1+(−7.78+5.65i)T+(16.3−50.4i)T2 1 + (-7.78 + 5.65i)T + (16.3 - 50.4i)T^{2}
59 1+(1.14−3.52i)T+(−47.7−34.6i)T2 1 + (1.14 - 3.52i)T + (-47.7 - 34.6i)T^{2}
61 1+(0.381+0.277i)T+(18.8+58.0i)T2 1 + (0.381 + 0.277i)T + (18.8 + 58.0i)T^{2}
67 1−0.381T+67T2 1 - 0.381T + 67T^{2}
71 1+(1.54+1.12i)T+(21.9+67.5i)T2 1 + (1.54 + 1.12i)T + (21.9 + 67.5i)T^{2}
73 1+(4.52−13.9i)T+(−59.0−42.9i)T2 1 + (4.52 - 13.9i)T + (-59.0 - 42.9i)T^{2}
79 1+(−4.92+3.57i)T+(24.4−75.1i)T2 1 + (-4.92 + 3.57i)T + (24.4 - 75.1i)T^{2}
83 1+(12.0+8.78i)T+(25.6+78.9i)T2 1 + (12.0 + 8.78i)T + (25.6 + 78.9i)T^{2}
89 1+6.76T+89T2 1 + 6.76T + 89T^{2}
97 1+(−12.4+9.06i)T+(29.9−92.2i)T2 1 + (-12.4 + 9.06i)T + (29.9 - 92.2i)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

−11.87187220332522575998880025122, −11.01539094545031490630140272195, −9.886776502821215447546746280384, −9.362921929705556450966508471645, −8.539841452101099138972822450480, −7.54010240415491691440529433101, −5.78750112228799382554800837272, −4.48246664919288151282031000848, −4.14025162933058475337135901628, −2.67938631635799050530584678710, 0.892262277878367080252750852680, 2.66396132816344527912412319426, 3.63870223819977626283936472493, 5.72970822304745179465034495623, 6.82010672324734605582715153679, 7.48235719840982172211784867013, 8.147584271986705331840373213454, 9.197861781217209399705654525591, 10.60798161907394083290085194552, 11.84127166563422548373774879425

Graph of the ZZ-function along the critical line