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Properties

Label 2-308-77.16-c1-0-3
Degree $2$
Conductor $308$
Sign $0.752 - 0.658i$
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  + (−0.0341 + 0.324i)3-s + (2.36 + 2.62i)5-s + (−0.532 − 2.59i)7-s + (2.83 + 0.601i)9-s + (−3.06 + 1.27i)11-s + (0.771 + 2.37i)13-s + (−0.933 + 0.678i)15-s + (−1.54 + 0.327i)17-s + (6.65 − 2.96i)19-s + (0.859 − 0.0846i)21-s + (−3.62 + 6.28i)23-s + (−0.782 + 7.44i)25-s + (−0.594 + 1.83i)27-s + (3.15 − 2.29i)29-s + (3.34 − 3.71i)31-s + ⋯
L(s)  = 1  + (−0.0197 + 0.187i)3-s + (1.05 + 1.17i)5-s + (−0.201 − 0.979i)7-s + (0.943 + 0.200i)9-s + (−0.923 + 0.383i)11-s + (0.213 + 0.658i)13-s + (−0.241 + 0.175i)15-s + (−0.373 + 0.0794i)17-s + (1.52 − 0.679i)19-s + (0.187 − 0.0184i)21-s + (−0.756 + 1.31i)23-s + (−0.156 + 1.48i)25-s + (−0.114 + 0.352i)27-s + (0.585 − 0.425i)29-s + (0.600 − 0.667i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.752 - 0.658i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42590 + 0.535522i\)
\(L(\frac12)\) \(\approx\) \(1.42590 + 0.535522i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.532 + 2.59i)T \)
11 \( 1 + (3.06 - 1.27i)T \)
good3 \( 1 + (0.0341 - 0.324i)T + (-2.93 - 0.623i)T^{2} \)
5 \( 1 + (-2.36 - 2.62i)T + (-0.522 + 4.97i)T^{2} \)
13 \( 1 + (-0.771 - 2.37i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (1.54 - 0.327i)T + (15.5 - 6.91i)T^{2} \)
19 \( 1 + (-6.65 + 2.96i)T + (12.7 - 14.1i)T^{2} \)
23 \( 1 + (3.62 - 6.28i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.15 + 2.29i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-3.34 + 3.71i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (0.638 + 6.07i)T + (-36.1 + 7.69i)T^{2} \)
41 \( 1 + (8.07 + 5.86i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.83T + 43T^{2} \)
47 \( 1 + (1.10 - 0.492i)T + (31.4 - 34.9i)T^{2} \)
53 \( 1 + (1.12 - 1.25i)T + (-5.54 - 52.7i)T^{2} \)
59 \( 1 + (4.70 + 2.09i)T + (39.4 + 43.8i)T^{2} \)
61 \( 1 + (6.27 + 6.96i)T + (-6.37 + 60.6i)T^{2} \)
67 \( 1 + (3.03 + 5.26i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-0.205 + 0.631i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.52 - 1.12i)T + (48.8 + 54.2i)T^{2} \)
79 \( 1 + (15.3 + 3.27i)T + (72.1 + 32.1i)T^{2} \)
83 \( 1 + (-1.50 + 4.61i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-5.86 + 10.1i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.58 + 17.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.55985813569956075404833473004, −10.62703193633571565521440823446, −10.01086196329682239724251466486, −9.450865127366265559182441878342, −7.56205945423772668562155904764, −7.08768290436697893998565952303, −5.99438115313032909177491049615, −4.68994648527795703518702040753, −3.34974725248926942998628244288, −1.92101196822121375080855296927, 1.35953071374729915641894913528, 2.85423440748634763554171795830, 4.77947954841924533144085394307, 5.55326153181507717149893638498, 6.46472504492246461335696172230, 8.028986089780057601547154902098, 8.734904814104571410116851676012, 9.795713490592039400931065431726, 10.31508482217885811202487501526, 11.96452729082446397941474354893

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