Properties

Label 2-338130-1.1-c1-0-6
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 + 2·7-s − 8-s + 10-s − 11-s − 13-s − 2·14-s + 16-s + 19-s − 20-s + 22-s + 25-s + 26-s + 2·28-s − 8·29-s − 5·31-s − 32-s − 2·35-s + 3·37-s − 38-s + 40-s + 4·41-s − 9·43-s − 44-s − 6·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s − 0.301·11-s − 0.277·13-s − 0.534·14-s + 1/4·16-s + 0.229·19-s − 0.223·20-s + 0.213·22-s + 1/5·25-s + 0.196·26-s + 0.377·28-s − 1.48·29-s − 0.898·31-s − 0.176·32-s − 0.338·35-s + 0.493·37-s − 0.162·38-s + 0.158·40-s + 0.624·41-s − 1.37·43-s − 0.150·44-s − 0.875·47-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

\(L(1)\) \(\approx\) \(0.6767039441\)
\(L(\frac12)\) \(\approx\) \(0.6767039441\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 - 4 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 3 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
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

−12.60519968558644, −11.85680658028199, −11.54596474377959, −11.30530596505546, −10.79037793514249, −10.29965570264753, −9.825579519215666, −9.388466333048565, −8.884520828157120, −8.418069396694778, −7.978133087025681, −7.579575362241645, −7.211088319735435, −6.703179321505957, −6.079698191251764, −5.471363821795323, −5.147497697408355, −4.533601794105322, −3.949465386673287, −3.410408497050896, −2.864266890224746, −2.129895124022902, −1.730904112096878, −1.089775507442350, −0.2551615891946762, 0.2551615891946762, 1.089775507442350, 1.730904112096878, 2.129895124022902, 2.864266890224746, 3.410408497050896, 3.949465386673287, 4.533601794105322, 5.147497697408355, 5.471363821795323, 6.079698191251764, 6.703179321505957, 7.211088319735435, 7.579575362241645, 7.978133087025681, 8.418069396694778, 8.884520828157120, 9.388466333048565, 9.825579519215666, 10.29965570264753, 10.79037793514249, 11.30530596505546, 11.54596474377959, 11.85680658028199, 12.60519968558644

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