Properties

Label 2-338-1.1-c7-0-27
Degree $2$
Conductor $338$
Sign $1$
Analytic cond. $105.586$
Root an. cond. $10.2755$
Motivic weight $7$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8·2-s − 27·3-s + 64·4-s + 245·5-s + 216·6-s + 587·7-s − 512·8-s − 1.45e3·9-s − 1.96e3·10-s + 3.87e3·11-s − 1.72e3·12-s − 4.69e3·14-s − 6.61e3·15-s + 4.09e3·16-s + 5.22e3·17-s + 1.16e4·18-s + 6.52e3·19-s + 1.56e4·20-s − 1.58e4·21-s − 3.09e4·22-s − 500·23-s + 1.38e4·24-s − 1.81e4·25-s + 9.84e4·27-s + 3.75e4·28-s + 2.26e5·29-s + 5.29e4·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.876·5-s + 0.408·6-s + 0.646·7-s − 0.353·8-s − 2/3·9-s − 0.619·10-s + 0.877·11-s − 0.288·12-s − 0.457·14-s − 0.506·15-s + 1/4·16-s + 0.258·17-s + 0.471·18-s + 0.218·19-s + 0.438·20-s − 0.373·21-s − 0.620·22-s − 0.00856·23-s + 0.204·24-s − 0.231·25-s + 0.962·27-s + 0.323·28-s + 1.72·29-s + 0.357·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(105.586\)
Root analytic conductor: \(10.2755\)
Motivic weight: \(7\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(1.668999165\)
\(L(\frac12)\) \(\approx\) \(1.668999165\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{3} T \)
13 \( 1 \)
good3 \( 1 + p^{3} T + p^{7} T^{2} \)
5 \( 1 - 49 p T + p^{7} T^{2} \)
7 \( 1 - 587 T + p^{7} T^{2} \)
11 \( 1 - 3874 T + p^{7} T^{2} \)
17 \( 1 - 5229 T + p^{7} T^{2} \)
19 \( 1 - 6522 T + p^{7} T^{2} \)
23 \( 1 + 500 T + p^{7} T^{2} \)
29 \( 1 - 7826 p T + p^{7} T^{2} \)
31 \( 1 + 130156 T + p^{7} T^{2} \)
37 \( 1 - 377769 T + p^{7} T^{2} \)
41 \( 1 - 539760 T + p^{7} T^{2} \)
43 \( 1 - 13987 T + p^{7} T^{2} \)
47 \( 1 - 526879 T + p^{7} T^{2} \)
53 \( 1 + 1649940 T + p^{7} T^{2} \)
59 \( 1 - 81194 T + p^{7} T^{2} \)
61 \( 1 + 1126952 T + p^{7} T^{2} \)
67 \( 1 + 478798 T + p^{7} T^{2} \)
71 \( 1 + 940007 T + p^{7} T^{2} \)
73 \( 1 + 1671926 T + p^{7} T^{2} \)
79 \( 1 + 5801188 T + p^{7} T^{2} \)
83 \( 1 + 7398816 T + p^{7} T^{2} \)
89 \( 1 - 953754 T + p^{7} T^{2} \)
97 \( 1 - 10318690 T + p^{7} T^{2} \)
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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

−10.30989791266988708152628551180, −9.418082421017030852542667046993, −8.620525383848856466477002674118, −7.58003523116581171390895124578, −6.33690288365007235691501641188, −5.77808302086069469612986506967, −4.57135215452421574024673137901, −2.90191319465376759810244836676, −1.68959915589195435923415352101, −0.73786693856012115675779994168, 0.73786693856012115675779994168, 1.68959915589195435923415352101, 2.90191319465376759810244836676, 4.57135215452421574024673137901, 5.77808302086069469612986506967, 6.33690288365007235691501641188, 7.58003523116581171390895124578, 8.620525383848856466477002674118, 9.418082421017030852542667046993, 10.30989791266988708152628551180

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