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Properties

Label 2-308-308.263-c1-0-36
Degree $2$
Conductor $308$
Sign $-0.323 + 0.946i$
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  + (1.39 − 0.228i)2-s + (−2.29 − 1.32i)3-s + (1.89 − 0.637i)4-s + (−3.5 − 1.32i)6-s + (0.5 − 2.59i)7-s + (2.49 − 1.32i)8-s + (2 + 3.46i)9-s + (−3.29 + 0.409i)11-s + (−5.18 − 1.04i)12-s − 2.64i·13-s + (0.104 − 3.74i)14-s + (3.18 − 2.41i)16-s + (−4.58 − 2.64i)17-s + (3.58 + 4.37i)18-s + (−4.58 + 5.29i)21-s + (−4.5 + 1.32i)22-s + ⋯
L(s)  = 1  + (0.986 − 0.161i)2-s + (−1.32 − 0.763i)3-s + (0.947 − 0.318i)4-s + (−1.42 − 0.540i)6-s + (0.188 − 0.981i)7-s + (0.883 − 0.467i)8-s + (0.666 + 1.15i)9-s + (−0.992 + 0.123i)11-s + (−1.49 − 0.302i)12-s − 0.733i·13-s + (0.0278 − 0.999i)14-s + (0.796 − 0.604i)16-s + (−1.11 − 0.641i)17-s + (0.844 + 1.03i)18-s + (−0.999 + 1.15i)21-s + (−0.959 + 0.282i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.323 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $-0.323 + 0.946i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ -0.323 + 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.855331 - 1.19575i\)
\(L(\frac12)\) \(\approx\) \(0.855331 - 1.19575i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.39 + 0.228i)T \)
7 \( 1 + (-0.5 + 2.59i)T \)
11 \( 1 + (3.29 - 0.409i)T \)
good3 \( 1 + (2.29 + 1.32i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
13 \( 1 + 2.64iT - 13T^{2} \)
17 \( 1 + (4.58 + 2.64i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.58 + 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.93iT - 29T^{2} \)
31 \( 1 + (-9.16 - 5.29i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4 - 6.92i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.29 - 1.32i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.29 + 1.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.29 - 1.32i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (-9.16 - 5.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−11.49624961309410699653524466650, −10.73767802608456938253703344607, −10.27698096812690365003583424580, −8.149163589166907966429394652271, −6.95883042136788658418464297997, −6.60143210499936082991329110987, −5.19063206548006622893456863828, −4.68596983327481833361181970022, −2.82184770882660956985849675235, −0.952084653986474754354346535006, 2.46110618836448759652781465770, 4.17924269116382637531296746780, 5.02621224051142506477212896199, 5.80909824950624064826701147276, 6.58875258456860559200844622730, 8.003166608613677448862360387990, 9.351299076186921942606459168651, 10.56368031646566724887172228355, 11.31944622142821509622600068676, 11.74909369496926883556991457824

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