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Properties

Label 2-308-11.9-c1-0-3
Degree $2$
Conductor $308$
Sign $0.957 - 0.288i$
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 − 0.726i)3-s + (0.618 + 1.90i)5-s + (0.809 + 0.587i)7-s + (−0.454 + 1.40i)9-s + (3.30 + 0.224i)11-s + (−0.381 + 1.17i)13-s + (2 + 1.45i)15-s + (−1.23 − 3.80i)17-s + (1.61 − 1.17i)19-s + 1.23·21-s − 1.61·23-s + (0.809 − 0.587i)25-s + (1.70 + 5.25i)27-s + (−4.73 − 3.44i)29-s + (1.38 − 4.25i)31-s + ⋯
L(s)  = 1  + (0.577 − 0.419i)3-s + (0.276 + 0.850i)5-s + (0.305 + 0.222i)7-s + (−0.151 + 0.466i)9-s + (0.997 + 0.0676i)11-s + (−0.105 + 0.326i)13-s + (0.516 + 0.375i)15-s + (−0.299 − 0.922i)17-s + (0.371 − 0.269i)19-s + 0.269·21-s − 0.337·23-s + (0.161 − 0.117i)25-s + (0.328 + 1.01i)27-s + (−0.879 − 0.638i)29-s + (0.248 − 0.763i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(308\)    =    \(2^{2} \cdot 7 \cdot 11\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(2.45939\)
Root analytic conductor: \(1.56824\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{308} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 308,\ (\ :1/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65691 + 0.243789i\)
\(L(\frac12)\) \(\approx\) \(1.65691 + 0.243789i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-3.30 - 0.224i)T \)
good3 \( 1 + (-1 + 0.726i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.618 - 1.90i)T + (-4.04 + 2.93i)T^{2} \)
13 \( 1 + (0.381 - 1.17i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (1.23 + 3.80i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-1.61 + 1.17i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 1.61T + 23T^{2} \)
29 \( 1 + (4.73 + 3.44i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-1.38 + 4.25i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.54 + 1.84i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.38 + 1.00i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 4.14T + 43T^{2} \)
47 \( 1 + (8.09 - 5.87i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (2.28 - 7.02i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (7.85 + 5.70i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.61 + 8.05i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.61T + 67T^{2} \)
71 \( 1 + (-4.04 - 12.4i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (13.4 + 9.78i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.57 + 4.84i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.909 + 2.80i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + (-3.52 + 10.8i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.57087416919543000906938075084, −11.00390761598988262054320470850, −9.716103549159145834698239182997, −8.963458669469558636055692239660, −7.79331464307134793872103870826, −7.02246682617515491617292738313, −5.98788465756155210273895720613, −4.56036230669541510202986083107, −3.02928327337187608891074917816, −1.95758593522394858933337205680, 1.46776686721088309497665475727, 3.38074203628068447641306600602, 4.36925146203061652264312814513, 5.59231429805122616705829787919, 6.76037859666680057048885816281, 8.162945899058982246129766692752, 8.884921664889740432602137449633, 9.568609284582158015878531893659, 10.59540264963289674413821966428, 11.76403075056413215079746043261

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