Properties

Label 2-350-1.1-c1-0-1
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 + 4-s + 7-s − 8-s − 3·9-s + 4·11-s + 6·13-s − 14-s + 16-s − 2·17-s + 3·18-s − 4·22-s − 6·26-s + 28-s + 6·29-s + 8·31-s − 32-s + 2·34-s − 3·36-s + 10·37-s + 2·41-s − 4·43-s + 4·44-s − 8·47-s + 49-s + 6·52-s + 2·53-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s + 0.377·7-s − 0.353·8-s − 9-s + 1.20·11-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.485·17-s + 0.707·18-s − 0.852·22-s − 1.17·26-s + 0.188·28-s + 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.342·34-s − 1/2·36-s + 1.64·37-s + 0.312·41-s − 0.609·43-s + 0.603·44-s − 1.16·47-s + 1/7·49-s + 0.832·52-s + 0.274·53-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

\(L(1)\) \(\approx\) \(1.055696301\)
\(L(\frac12)\) \(\approx\) \(1.055696301\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
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

−11.41432333521757255890378761766, −10.68041607351861279913180531310, −9.460094722609108544443052630434, −8.629572853345107589851782221455, −8.089729535357517933440155073370, −6.56718920518778472538358027386, −6.01430128602422585899125610915, −4.36823223624486251008730105784, −2.98711940319117381089884417433, −1.27008084899723889248060652436, 1.27008084899723889248060652436, 2.98711940319117381089884417433, 4.36823223624486251008730105784, 6.01430128602422585899125610915, 6.56718920518778472538358027386, 8.089729535357517933440155073370, 8.629572853345107589851782221455, 9.460094722609108544443052630434, 10.68041607351861279913180531310, 11.41432333521757255890378761766

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