Properties

Label 2-338130-1.1-c1-0-10
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 + 4·11-s + 13-s + 2·14-s + 16-s + 2·19-s − 20-s − 4·22-s − 8·23-s + 25-s − 26-s − 2·28-s − 2·29-s − 9·31-s − 32-s + 2·35-s − 6·37-s − 2·38-s + 40-s − 6·41-s + 4·43-s + 4·44-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 + 1.20·11-s + 0.277·13-s + 0.534·14-s + 1/4·16-s + 0.458·19-s − 0.223·20-s − 0.852·22-s − 1.66·23-s + 1/5·25-s − 0.196·26-s − 0.377·28-s − 0.371·29-s − 1.61·31-s − 0.176·32-s + 0.338·35-s − 0.986·37-s − 0.324·38-s + 0.158·40-s − 0.937·41-s + 0.609·43-s + 0.603·44-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

\(L(1)\) \(\approx\) \(0.7854750148\)
\(L(\frac12)\) \(\approx\) \(0.7854750148\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 - 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 3 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 9 T + p T^{2} \)
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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

−12.34656076142318, −12.06268402930688, −11.77157368164671, −11.12598219265047, −10.78315621396569, −10.23589243265547, −9.791944076134050, −9.326010724490858, −9.000450860943125, −8.558853135357003, −7.984173734101548, −7.511546967277384, −7.061187507293026, −6.648475564816154, −6.118711460502798, −5.729973358953938, −5.145289184255771, −4.345404101532140, −3.722262011405161, −3.622876939416442, −2.979763093691466, −2.072900695148416, −1.772349119688979, −0.9958474465718613, −0.2934165226744504, 0.2934165226744504, 0.9958474465718613, 1.772349119688979, 2.072900695148416, 2.979763093691466, 3.622876939416442, 3.722262011405161, 4.345404101532140, 5.145289184255771, 5.729973358953938, 6.118711460502798, 6.648475564816154, 7.061187507293026, 7.511546967277384, 7.984173734101548, 8.558853135357003, 9.000450860943125, 9.326010724490858, 9.791944076134050, 10.23589243265547, 10.78315621396569, 11.12598219265047, 11.77157368164671, 12.06268402930688, 12.34656076142318

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