Properties

Label 2-350-1.1-c1-0-5
Degree $2$
Conductor $350$
Sign $1$
Analytic cond. $2.79476$
Root an. cond. $1.67175$
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 + 3-s + 4-s + 6-s + 7-s + 8-s − 2·9-s + 3·11-s + 12-s + 2·13-s + 14-s + 16-s + 3·17-s − 2·18-s − 7·19-s + 21-s + 3·22-s + 24-s + 2·26-s − 5·27-s + 28-s − 6·29-s − 4·31-s + 32-s + 3·33-s + 3·34-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s + 0.353·8-s − 2/3·9-s + 0.904·11-s + 0.288·12-s + 0.554·13-s + 0.267·14-s + 1/4·16-s + 0.727·17-s − 0.471·18-s − 1.60·19-s + 0.218·21-s + 0.639·22-s + 0.204·24-s + 0.392·26-s − 0.962·27-s + 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + 0.514·34-s − 1/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(350\)    =    \(2 \cdot 5^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(2.79476\)
Root analytic conductor: \(1.67175\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.403179303\)
\(L(\frac12)\) \(\approx\) \(2.403179303\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 7 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 7 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 5 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−11.45909715795481371039597944171, −10.89274588760446240115568394507, −9.536378818315516005204945079997, −8.610022790516695549795853061214, −7.77658810173983217795932841222, −6.50150023662226984648693008298, −5.63302791327341414299631153495, −4.26850617197803948832200761616, −3.32408442847114247075118763103, −1.89984331795161626562592295843, 1.89984331795161626562592295843, 3.32408442847114247075118763103, 4.26850617197803948832200761616, 5.63302791327341414299631153495, 6.50150023662226984648693008298, 7.77658810173983217795932841222, 8.610022790516695549795853061214, 9.536378818315516005204945079997, 10.89274588760446240115568394507, 11.45909715795481371039597944171

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