Properties

Label 2-338-1.1-c9-0-29
Degree $2$
Conductor $338$
Sign $1$
Analytic cond. $174.082$
Root an. cond. $13.1940$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 16·2-s − 273·3-s + 256·4-s − 1.01e3·5-s − 4.36e3·6-s − 3.95e3·7-s + 4.09e3·8-s + 5.48e4·9-s − 1.62e4·10-s + 5.09e4·11-s − 6.98e4·12-s − 6.32e4·14-s + 2.77e5·15-s + 6.55e4·16-s + 5.09e5·17-s + 8.77e5·18-s + 6.26e5·19-s − 2.59e5·20-s + 1.07e6·21-s + 8.15e5·22-s + 6.53e5·23-s − 1.11e6·24-s − 9.22e5·25-s − 9.59e6·27-s − 1.01e6·28-s − 4.94e6·29-s + 4.43e6·30-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.94·3-s + 1/2·4-s − 0.726·5-s − 1.37·6-s − 0.622·7-s + 0.353·8-s + 2.78·9-s − 0.513·10-s + 1.05·11-s − 0.972·12-s − 0.440·14-s + 1.41·15-s + 1/4·16-s + 1.48·17-s + 1.97·18-s + 1.10·19-s − 0.363·20-s + 1.21·21-s + 0.742·22-s + 0.486·23-s − 0.687·24-s − 0.472·25-s − 3.47·27-s − 0.311·28-s − 1.29·29-s + 0.999·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(174.082\)
Root analytic conductor: \(13.1940\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :9/2),\ 1)\)

Particular Values

\(L(5)\) \(\approx\) \(1.509486585\)
\(L(\frac12)\) \(\approx\) \(1.509486585\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{4} T \)
13 \( 1 \)
good3 \( 1 + 91 p T + p^{9} T^{2} \)
5 \( 1 + 203 p T + p^{9} T^{2} \)
7 \( 1 + 565 p T + p^{9} T^{2} \)
11 \( 1 - 50998 T + p^{9} T^{2} \)
17 \( 1 - 509757 T + p^{9} T^{2} \)
19 \( 1 - 626574 T + p^{9} T^{2} \)
23 \( 1 - 653524 T + p^{9} T^{2} \)
29 \( 1 + 4943006 T + p^{9} T^{2} \)
31 \( 1 + 4071700 T + p^{9} T^{2} \)
37 \( 1 + 2348883 T + p^{9} T^{2} \)
41 \( 1 - 13350960 T + p^{9} T^{2} \)
43 \( 1 + 7834847 T + p^{9} T^{2} \)
47 \( 1 - 39637681 T + p^{9} T^{2} \)
53 \( 1 - 73200924 T + p^{9} T^{2} \)
59 \( 1 - 141141614 T + p^{9} T^{2} \)
61 \( 1 + 132061256 T + p^{9} T^{2} \)
67 \( 1 - 185673110 T + p^{9} T^{2} \)
71 \( 1 + 224452625 T + p^{9} T^{2} \)
73 \( 1 - 2363338 p T + p^{9} T^{2} \)
79 \( 1 + 643288156 T + p^{9} T^{2} \)
83 \( 1 + 720077280 T + p^{9} T^{2} \)
89 \( 1 - 73028106 T + p^{9} T^{2} \)
97 \( 1 - 15879778 T + p^{9} T^{2} \)
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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

−10.26816822972103868555994121455, −9.455000979266892719060184805722, −7.48716594715266118519925592368, −6.99929639864244902410519257413, −5.85847312183017710475032189868, −5.41310590195499113512283772360, −4.16400916500400204301550399584, −3.50271629809486983932378229789, −1.44114462605644691857643339500, −0.57674611145920558468241113567, 0.57674611145920558468241113567, 1.44114462605644691857643339500, 3.50271629809486983932378229789, 4.16400916500400204301550399584, 5.41310590195499113512283772360, 5.85847312183017710475032189868, 6.99929639864244902410519257413, 7.48716594715266118519925592368, 9.455000979266892719060184805722, 10.26816822972103868555994121455

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