Properties

Label 2-338130-1.1-c1-0-9
Degree $2$
Conductor $338130$
Sign $1$
Analytic cond. $2699.98$
Root an. cond. $51.9613$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 5-s + 3·7-s − 8-s + 10-s + 2·11-s − 13-s − 3·14-s + 16-s − 2·19-s − 20-s − 2·22-s − 3·23-s + 25-s + 26-s + 3·28-s + 2·29-s − 32-s − 3·35-s − 2·37-s + 2·38-s + 40-s + 5·41-s + 4·43-s + 2·44-s + 3·46-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.13·7-s − 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.277·13-s − 0.801·14-s + 1/4·16-s − 0.458·19-s − 0.223·20-s − 0.426·22-s − 0.625·23-s + 1/5·25-s + 0.196·26-s + 0.566·28-s + 0.371·29-s − 0.176·32-s − 0.507·35-s − 0.328·37-s + 0.324·38-s + 0.158·40-s + 0.780·41-s + 0.609·43-s + 0.301·44-s + 0.442·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338130\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 13 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2699.98\)
Root analytic conductor: \(51.9613\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{338130} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338130,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9474628590\)
\(L(\frac12)\) \(\approx\) \(0.9474628590\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + T \)
17 \( 1 \)
good7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−12.49577268260770, −11.92542669949492, −11.62041700736200, −11.28072321707321, −10.67588111139675, −10.42319417044242, −9.870746289737036, −9.250412050562012, −8.874003351614195, −8.529101702613312, −7.865714241050323, −7.672588333190890, −7.252722804535652, −6.540091777823667, −6.153238650185790, −5.656061351252449, −4.928515881025500, −4.404287925192931, −4.225180023119502, −3.303024871466348, −2.907732620829655, −2.132867527948662, −1.591351945020409, −1.220342872038547, −0.2906000509383493, 0.2906000509383493, 1.220342872038547, 1.591351945020409, 2.132867527948662, 2.907732620829655, 3.303024871466348, 4.225180023119502, 4.404287925192931, 4.928515881025500, 5.656061351252449, 6.153238650185790, 6.540091777823667, 7.252722804535652, 7.672588333190890, 7.865714241050323, 8.529101702613312, 8.874003351614195, 9.250412050562012, 9.870746289737036, 10.42319417044242, 10.67588111139675, 11.28072321707321, 11.62041700736200, 11.92542669949492, 12.49577268260770

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