Properties

Label 2-350-1.1-c1-0-9
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3·3-s + 4-s − 3·6-s − 7-s + 8-s + 6·9-s − 5·11-s − 3·12-s − 6·13-s − 14-s + 16-s − 17-s + 6·18-s − 3·19-s + 3·21-s − 5·22-s − 3·24-s − 6·26-s − 9·27-s − 28-s − 6·29-s − 4·31-s + 32-s + 15·33-s − 34-s + 6·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s − 0.377·7-s + 0.353·8-s + 2·9-s − 1.50·11-s − 0.866·12-s − 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.41·18-s − 0.688·19-s + 0.654·21-s − 1.06·22-s − 0.612·24-s − 1.17·26-s − 1.73·27-s − 0.188·28-s − 1.11·29-s − 0.718·31-s + 0.176·32-s + 2.61·33-s − 0.171·34-s + 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: \(1\)
Selberg data: \((2,\ 350,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
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 \( 1 - 11 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 9 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 7 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
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

−11.11858258197447904087693203763, −10.45494407305386764464659959764, −9.627920882288908715420031969238, −7.70054391482804408966188304160, −6.95538626651508714624591948210, −5.84573096199937208498827543184, −5.20618442524944639616146822933, −4.32124854619714589271574708665, −2.44905781159727834481891285350, 0, 2.44905781159727834481891285350, 4.32124854619714589271574708665, 5.20618442524944639616146822933, 5.84573096199937208498827543184, 6.95538626651508714624591948210, 7.70054391482804408966188304160, 9.627920882288908715420031969238, 10.45494407305386764464659959764, 11.11858258197447904087693203763

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