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Properties

Label 2-308-28.19-c1-0-15
Degree $2$
Conductor $308$
Sign $0.929 - 0.367i$
Analytic cond. $2.45939$
Root an. cond. $1.56824$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.18 − 0.766i)2-s + (0.968 + 1.67i)3-s + (0.823 + 1.82i)4-s + (2.62 + 1.51i)5-s + (0.135 − 2.73i)6-s + (1.13 − 2.38i)7-s + (0.419 − 2.79i)8-s + (−0.376 + 0.651i)9-s + (−1.95 − 3.81i)10-s + (0.866 − 0.5i)11-s + (−2.26 + 3.14i)12-s − 1.21i·13-s + (−3.18 + 1.96i)14-s + 5.87i·15-s + (−2.64 + 3.00i)16-s + (2.50 − 1.44i)17-s + ⋯
L(s)  = 1  + (−0.840 − 0.542i)2-s + (0.559 + 0.968i)3-s + (0.411 + 0.911i)4-s + (1.17 + 0.678i)5-s + (0.0554 − 1.11i)6-s + (0.430 − 0.902i)7-s + (0.148 − 0.988i)8-s + (−0.125 + 0.217i)9-s + (−0.618 − 1.20i)10-s + (0.261 − 0.150i)11-s + (−0.652 + 0.908i)12-s − 0.335i·13-s + (−0.850 + 0.525i)14-s + 1.51i·15-s + (−0.660 + 0.750i)16-s + (0.607 − 0.350i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.929 - 0.367i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.929 - 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30524 + 0.248689i\)
\(L(\frac12)\) \(\approx\) \(1.30524 + 0.248689i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 + 0.766i)T \)
7 \( 1 + (-1.13 + 2.38i)T \)
11 \( 1 + (-0.866 + 0.5i)T \)
good3 \( 1 + (-0.968 - 1.67i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.62 - 1.51i)T + (2.5 + 4.33i)T^{2} \)
13 \( 1 + 1.21iT - 13T^{2} \)
17 \( 1 + (-2.50 + 1.44i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.58 - 6.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.33 + 0.768i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 9.30T + 29T^{2} \)
31 \( 1 + (1.61 + 2.80i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.02 - 5.24i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 9.31iT - 41T^{2} \)
43 \( 1 - 9.31iT - 43T^{2} \)
47 \( 1 + (-0.0158 + 0.0275i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.0965 - 0.167i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.813 - 1.40i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.93 - 1.69i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.71 - 3.87i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 + (13.1 - 7.56i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.49 - 4.90i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 1.01T + 83T^{2} \)
89 \( 1 + (3.85 + 2.22i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.53iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.29134469246314297012726196543, −10.35150291225580852678877229781, −10.08663643169264643345717804659, −9.267422115385409761600182957918, −8.199392493836157379295419534415, −7.16059036966647735179105132056, −5.91765773464785323713987214717, −4.14529502943565256586463932320, −3.23803146654408365365312645654, −1.77803628375319741805896985552, 1.59282156773612795585715941884, 2.26699440898084129752418352087, 5.02855519975042680906124587049, 5.91364169938030762112747076969, 6.92204216099679144920438444959, 7.928303075452501995272802709788, 8.944493871801249534205433397914, 9.206868750052007275673062184938, 10.47648965146845127874352536256, 11.64120667082966937295439655691

Graph of the $Z$-function along the critical line