Properties

Label 2-338130-1.1-c1-0-1
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 − 4·7-s − 8-s + 10-s − 5·11-s − 13-s + 4·14-s + 16-s + 19-s − 20-s + 5·22-s + 3·23-s + 25-s + 26-s − 4·28-s − 9·29-s + 3·31-s − 32-s + 4·35-s − 37-s − 38-s + 40-s − 8·41-s + 11·43-s − 5·44-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s − 1.51·7-s − 0.353·8-s + 0.316·10-s − 1.50·11-s − 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s + 1.06·22-s + 0.625·23-s + 1/5·25-s + 0.196·26-s − 0.755·28-s − 1.67·29-s + 0.538·31-s − 0.176·32-s + 0.676·35-s − 0.164·37-s − 0.162·38-s + 0.158·40-s − 1.24·41-s + 1.67·43-s − 0.753·44-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

\(L(1)\) \(\approx\) \(0.1200222644\)
\(L(\frac12)\) \(\approx\) \(0.1200222644\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 15 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 3 T + p T^{2} \)
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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

−12.52929367148121, −12.17704184806761, −11.67203953062434, −10.98551654543683, −10.78493793163726, −10.15903017998703, −9.945195908361203, −9.397873020965811, −8.932832174600466, −8.580776755487546, −7.874469091753488, −7.499036089533192, −7.148890929426537, −6.737784007312125, −6.012846350514916, −5.641386124556549, −5.221822106138435, −4.442914228575191, −3.899566752922714, −3.252202118023885, −2.853419230037617, −2.490870455563920, −1.705638946359795, −0.8587281010357263, −0.1253518978050891, 0.1253518978050891, 0.8587281010357263, 1.705638946359795, 2.490870455563920, 2.853419230037617, 3.252202118023885, 3.899566752922714, 4.442914228575191, 5.221822106138435, 5.641386124556549, 6.012846350514916, 6.737784007312125, 7.148890929426537, 7.499036089533192, 7.874469091753488, 8.580776755487546, 8.932832174600466, 9.397873020965811, 9.945195908361203, 10.15903017998703, 10.78493793163726, 10.98551654543683, 11.67203953062434, 12.17704184806761, 12.52929367148121

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