Properties

Label 2-338-1.1-c3-0-14
Degree $2$
Conductor $338$
Sign $1$
Analytic cond. $19.9426$
Root an. cond. $4.46571$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 4·3-s + 4·4-s + 18·5-s − 8·6-s − 20·7-s − 8·8-s − 11·9-s − 36·10-s + 48·11-s + 16·12-s + 40·14-s + 72·15-s + 16·16-s + 66·17-s + 22·18-s + 16·19-s + 72·20-s − 80·21-s − 96·22-s + 168·23-s − 32·24-s + 199·25-s − 152·27-s − 80·28-s + 6·29-s − 144·30-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.769·3-s + 1/2·4-s + 1.60·5-s − 0.544·6-s − 1.07·7-s − 0.353·8-s − 0.407·9-s − 1.13·10-s + 1.31·11-s + 0.384·12-s + 0.763·14-s + 1.23·15-s + 1/4·16-s + 0.941·17-s + 0.288·18-s + 0.193·19-s + 0.804·20-s − 0.831·21-s − 0.930·22-s + 1.52·23-s − 0.272·24-s + 1.59·25-s − 1.08·27-s − 0.539·28-s + 0.0384·29-s − 0.876·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.233703938\)
\(L(\frac12)\) \(\approx\) \(2.233703938\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p T \)
13 \( 1 \)
good3 \( 1 - 4 T + p^{3} T^{2} \)
5 \( 1 - 18 T + p^{3} T^{2} \)
7 \( 1 + 20 T + p^{3} T^{2} \)
11 \( 1 - 48 T + p^{3} T^{2} \)
17 \( 1 - 66 T + p^{3} T^{2} \)
19 \( 1 - 16 T + p^{3} T^{2} \)
23 \( 1 - 168 T + p^{3} T^{2} \)
29 \( 1 - 6 T + p^{3} T^{2} \)
31 \( 1 + 20 T + p^{3} T^{2} \)
37 \( 1 + 254 T + p^{3} T^{2} \)
41 \( 1 - 390 T + p^{3} T^{2} \)
43 \( 1 + 124 T + p^{3} T^{2} \)
47 \( 1 - 468 T + p^{3} T^{2} \)
53 \( 1 - 558 T + p^{3} T^{2} \)
59 \( 1 - 96 T + p^{3} T^{2} \)
61 \( 1 + 826 T + p^{3} T^{2} \)
67 \( 1 - 160 T + p^{3} T^{2} \)
71 \( 1 - 420 T + p^{3} T^{2} \)
73 \( 1 + 362 T + p^{3} T^{2} \)
79 \( 1 - 776 T + p^{3} T^{2} \)
83 \( 1 + p^{3} T^{2} \)
89 \( 1 + 1626 T + p^{3} T^{2} \)
97 \( 1 - 1294 T + p^{3} T^{2} \)
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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

−10.76820273372700194898543517619, −9.812442177620602446001182782847, −9.237161619722184036797842416135, −8.797185775220281037758790096253, −7.26890135200266772804228937012, −6.35340326888941469001705118611, −5.55049645615271480416844436735, −3.45700288523611366047200292298, −2.50858003971681169142693028567, −1.18052842001643421970454353344, 1.18052842001643421970454353344, 2.50858003971681169142693028567, 3.45700288523611366047200292298, 5.55049645615271480416844436735, 6.35340326888941469001705118611, 7.26890135200266772804228937012, 8.797185775220281037758790096253, 9.237161619722184036797842416135, 9.812442177620602446001182782847, 10.76820273372700194898543517619

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