Properties

Label 2-338-1.1-c5-0-9
Degree $2$
Conductor $338$
Sign $1$
Analytic cond. $54.2097$
Root an. cond. $7.36272$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s − 13·3-s + 16·4-s + 51·5-s + 52·6-s − 105·7-s − 64·8-s − 74·9-s − 204·10-s − 120·11-s − 208·12-s + 420·14-s − 663·15-s + 256·16-s + 1.10e3·17-s + 296·18-s − 1.17e3·19-s + 816·20-s + 1.36e3·21-s + 480·22-s − 1.05e3·23-s + 832·24-s − 524·25-s + 4.12e3·27-s − 1.68e3·28-s − 4.10e3·29-s + 2.65e3·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.833·3-s + 1/2·4-s + 0.912·5-s + 0.589·6-s − 0.809·7-s − 0.353·8-s − 0.304·9-s − 0.645·10-s − 0.299·11-s − 0.416·12-s + 0.572·14-s − 0.760·15-s + 1/4·16-s + 0.923·17-s + 0.215·18-s − 0.743·19-s + 0.456·20-s + 0.675·21-s + 0.211·22-s − 0.413·23-s + 0.294·24-s − 0.167·25-s + 1.08·27-s − 0.404·28-s − 0.906·29-s + 0.537·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(54.2097\)
Root analytic conductor: \(7.36272\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(0.7512904379\)
\(L(\frac12)\) \(\approx\) \(0.7512904379\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
13 \( 1 \)
good3 \( 1 + 13 T + p^{5} T^{2} \)
5 \( 1 - 51 T + p^{5} T^{2} \)
7 \( 1 + 15 p T + p^{5} T^{2} \)
11 \( 1 + 120 T + p^{5} T^{2} \)
17 \( 1 - 1101 T + p^{5} T^{2} \)
19 \( 1 + 1170 T + p^{5} T^{2} \)
23 \( 1 + 1050 T + p^{5} T^{2} \)
29 \( 1 + 4104 T + p^{5} T^{2} \)
31 \( 1 - 9624 T + p^{5} T^{2} \)
37 \( 1 + 8709 T + p^{5} T^{2} \)
41 \( 1 + 9480 T + p^{5} T^{2} \)
43 \( 1 + 9995 T + p^{5} T^{2} \)
47 \( 1 - 2943 T + p^{5} T^{2} \)
53 \( 1 + 750 T + p^{5} T^{2} \)
59 \( 1 - 40938 T + p^{5} T^{2} \)
61 \( 1 + 57920 T + p^{5} T^{2} \)
67 \( 1 - 22812 T + p^{5} T^{2} \)
71 \( 1 - 63741 T + p^{5} T^{2} \)
73 \( 1 + 58866 T + p^{5} T^{2} \)
79 \( 1 - 63202 T + p^{5} T^{2} \)
83 \( 1 - 55458 T + p^{5} T^{2} \)
89 \( 1 - 104778 T + p^{5} T^{2} \)
97 \( 1 - 160452 T + p^{5} T^{2} \)
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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

−10.36714334787932249958816362393, −10.04656518616522992025409937744, −8.998491853789444426641052915033, −7.960036575021145833544666426895, −6.60182321900709373038801262211, −6.05730641755049381431731746161, −5.12300298452025763369030226381, −3.28367568947472807847285697873, −1.99487787273563304465686157502, −0.53290104551379323844062942223, 0.53290104551379323844062942223, 1.99487787273563304465686157502, 3.28367568947472807847285697873, 5.12300298452025763369030226381, 6.05730641755049381431731746161, 6.60182321900709373038801262211, 7.960036575021145833544666426895, 8.998491853789444426641052915033, 10.04656518616522992025409937744, 10.36714334787932249958816362393

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