Properties

Label 2-304-304.149-c1-0-24
Degree $2$
Conductor $304$
Sign $0.128 - 0.991i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
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  + (1.02 + 0.971i)2-s + (−0.719 + 1.54i)3-s + (0.110 + 1.99i)4-s + (3.48 − 2.43i)5-s + (−2.23 + 0.885i)6-s + (1.42 − 0.824i)7-s + (−1.82 + 2.15i)8-s + (0.0677 + 0.0807i)9-s + (5.95 + 0.880i)10-s + (1.16 − 0.311i)11-s + (−3.15 − 1.26i)12-s + (−1.15 + 0.537i)13-s + (2.26 + 0.540i)14-s + (1.25 + 7.12i)15-s + (−3.97 + 0.442i)16-s + (−5.18 − 4.34i)17-s + ⋯
L(s)  = 1  + (0.726 + 0.687i)2-s + (−0.415 + 0.890i)3-s + (0.0553 + 0.998i)4-s + (1.55 − 1.09i)5-s + (−0.913 + 0.361i)6-s + (0.539 − 0.311i)7-s + (−0.645 + 0.763i)8-s + (0.0225 + 0.0269i)9-s + (1.88 + 0.278i)10-s + (0.350 − 0.0939i)11-s + (−0.911 − 0.365i)12-s + (−0.319 + 0.149i)13-s + (0.606 + 0.144i)14-s + (0.324 + 1.84i)15-s + (−0.993 + 0.110i)16-s + (−1.25 − 1.05i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $0.128 - 0.991i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ 0.128 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55596 + 1.36796i\)
\(L(\frac12)\) \(\approx\) \(1.55596 + 1.36796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.02 - 0.971i)T \)
19 \( 1 + (4.31 - 0.600i)T \)
good3 \( 1 + (0.719 - 1.54i)T + (-1.92 - 2.29i)T^{2} \)
5 \( 1 + (-3.48 + 2.43i)T + (1.71 - 4.69i)T^{2} \)
7 \( 1 + (-1.42 + 0.824i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.16 + 0.311i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.15 - 0.537i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (5.18 + 4.34i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (4.57 - 0.806i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (-0.370 + 4.23i)T + (-28.5 - 5.03i)T^{2} \)
31 \( 1 + (-4.45 - 7.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.56 - 1.56i)T + 37iT^{2} \)
41 \( 1 + (-1.99 - 5.48i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (-5.82 + 4.08i)T + (14.7 - 40.4i)T^{2} \)
47 \( 1 + (0.111 - 0.0939i)T + (8.16 - 46.2i)T^{2} \)
53 \( 1 + (-4.94 + 7.06i)T + (-18.1 - 49.8i)T^{2} \)
59 \( 1 + (0.363 + 4.14i)T + (-58.1 + 10.2i)T^{2} \)
61 \( 1 + (4.07 + 2.85i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (-0.492 + 5.63i)T + (-65.9 - 11.6i)T^{2} \)
71 \( 1 + (4.68 + 0.826i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.61 - 7.19i)T + (-55.9 + 46.9i)T^{2} \)
79 \( 1 + (2.48 - 0.903i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (4.58 + 1.22i)T + (71.8 + 41.5i)T^{2} \)
89 \( 1 + (4.51 - 12.4i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (1.60 + 1.35i)T + (16.8 + 95.5i)T^{2} \)
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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

−12.09561264544053900083888489741, −11.03073069383206891800435890466, −9.933355849363165582092872086924, −9.139456041551609873420440270614, −8.214231537527947466736297277817, −6.68548307851681260587967842724, −5.76134440836463554556175458458, −4.75923234119403769203816578234, −4.41212314735911451810182159486, −2.17245901493586784845873503042, 1.77475754086134724451273722488, 2.43313877974350257264174040903, 4.27164666158853735667934988901, 5.91204212799256373492827908841, 6.18544416124672714434230944461, 7.14464953199095751809903763160, 8.936292224313942114336050898520, 10.00490393939495608763004887975, 10.73560713324613813408520806398, 11.52124673928312722662277276055

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