Properties

Label 2-304-16.13-c1-0-7
Degree $2$
Conductor $304$
Sign $-0.843 - 0.536i$
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  + (0.484 + 1.32i)2-s + (0.504 + 0.504i)3-s + (−1.53 + 1.28i)4-s + (−0.312 + 0.312i)5-s + (−0.425 + 0.913i)6-s + 4.86i·7-s + (−2.45 − 1.41i)8-s − 2.49i·9-s + (−0.566 − 0.263i)10-s + (−0.0288 + 0.0288i)11-s + (−1.42 − 0.123i)12-s + (0.751 + 0.751i)13-s + (−6.47 + 2.35i)14-s − 0.314·15-s + (0.690 − 3.93i)16-s − 2.20·17-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)2-s + (0.291 + 0.291i)3-s + (−0.765 + 0.643i)4-s + (−0.139 + 0.139i)5-s + (−0.173 + 0.373i)6-s + 1.84i·7-s + (−0.866 − 0.499i)8-s − 0.830i·9-s + (−0.179 − 0.0834i)10-s + (−0.00871 + 0.00871i)11-s + (−0.410 − 0.0356i)12-s + (0.208 + 0.208i)13-s + (−1.72 + 0.629i)14-s − 0.0813·15-s + (0.172 − 0.984i)16-s − 0.535·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.843 - 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.843 - 0.536i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.843 - 0.536i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.394081 + 1.35460i\)
\(L(\frac12)\) \(\approx\) \(0.394081 + 1.35460i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.484 - 1.32i)T \)
19 \( 1 + (-0.707 - 0.707i)T \)
good3 \( 1 + (-0.504 - 0.504i)T + 3iT^{2} \)
5 \( 1 + (0.312 - 0.312i)T - 5iT^{2} \)
7 \( 1 - 4.86iT - 7T^{2} \)
11 \( 1 + (0.0288 - 0.0288i)T - 11iT^{2} \)
13 \( 1 + (-0.751 - 0.751i)T + 13iT^{2} \)
17 \( 1 + 2.20T + 17T^{2} \)
23 \( 1 - 2.17iT - 23T^{2} \)
29 \( 1 + (-1.26 - 1.26i)T + 29iT^{2} \)
31 \( 1 - 5.96T + 31T^{2} \)
37 \( 1 + (-0.874 + 0.874i)T - 37iT^{2} \)
41 \( 1 + 2.77iT - 41T^{2} \)
43 \( 1 + (-6.46 + 6.46i)T - 43iT^{2} \)
47 \( 1 - 5.54T + 47T^{2} \)
53 \( 1 + (-8.87 + 8.87i)T - 53iT^{2} \)
59 \( 1 + (3.76 - 3.76i)T - 59iT^{2} \)
61 \( 1 + (-0.891 - 0.891i)T + 61iT^{2} \)
67 \( 1 + (-9.15 - 9.15i)T + 67iT^{2} \)
71 \( 1 - 1.49iT - 71T^{2} \)
73 \( 1 - 1.57iT - 73T^{2} \)
79 \( 1 + 8.59T + 79T^{2} \)
83 \( 1 + (2.79 + 2.79i)T + 83iT^{2} \)
89 \( 1 + 11.9iT - 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−12.17022094701760159547097777785, −11.54468839013466629307597263601, −9.812188314583585568411006246144, −8.925422474348377188468138491857, −8.548277634953952732700992929675, −7.14626536454232535483168679286, −6.09666960795333598040840111221, −5.32745414836828167857269824160, −3.94910193890330893553176555757, −2.75000320190210766115096488200, 0.975494812743305049380696050722, 2.63586389577523752175871724885, 4.07376169955435770879238733677, 4.76550253259956660375681740141, 6.40312449714670108038743436477, 7.62557547557335523426138950716, 8.471646391684067149084361162981, 9.816012644024541307397464805321, 10.60392082405646970064600322039, 11.12578714273138759208618840774

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