Properties

Label 2-350-1.1-c1-0-6
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 + 2·3-s + 4-s + 2·6-s − 7-s + 8-s + 9-s + 2·12-s + 4·13-s − 14-s + 16-s − 6·17-s + 18-s + 2·19-s − 2·21-s + 2·24-s + 4·26-s − 4·27-s − 28-s − 6·29-s − 4·31-s + 32-s − 6·34-s + 36-s − 2·37-s + 2·38-s + 8·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.458·19-s − 0.436·21-s + 0.408·24-s + 0.784·26-s − 0.769·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.324·38-s + 1.28·39-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

\(L(1)\) \(\approx\) \(2.658249180\)
\(L(\frac12)\) \(\approx\) \(2.658249180\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−11.46841286601878794911833201781, −10.74873683533699837976941407431, −9.395997320196623402140164716317, −8.758164081717971221129539824522, −7.71895664384636679391588711627, −6.66965130292041403123447657278, −5.59857578974307577358363282055, −4.11500666640753372808858013263, −3.28327332646103144207952017478, −2.07204529193989626181377807359, 2.07204529193989626181377807359, 3.28327332646103144207952017478, 4.11500666640753372808858013263, 5.59857578974307577358363282055, 6.66965130292041403123447657278, 7.71895664384636679391588711627, 8.758164081717971221129539824522, 9.395997320196623402140164716317, 10.74873683533699837976941407431, 11.46841286601878794911833201781

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