Properties

Label 2-650-1.1-c1-0-17
Degree $2$
Conductor $650$
Sign $-1$
Analytic cond. $5.19027$
Root an. cond. $2.27821$
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-s + 4-s − 6-s − 4·7-s + 8-s − 2·9-s + 11-s − 12-s − 13-s − 4·14-s + 16-s − 7·17-s − 2·18-s − 3·19-s + 4·21-s + 22-s − 24-s − 26-s + 5·27-s − 4·28-s − 4·29-s + 6·31-s + 32-s − 33-s − 7·34-s − 2·36-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.301·11-s − 0.288·12-s − 0.277·13-s − 1.06·14-s + 1/4·16-s − 1.69·17-s − 0.471·18-s − 0.688·19-s + 0.872·21-s + 0.213·22-s − 0.204·24-s − 0.196·26-s + 0.962·27-s − 0.755·28-s − 0.742·29-s + 1.07·31-s + 0.176·32-s − 0.174·33-s − 1.20·34-s − 1/3·36-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(5.19027\)
Root analytic conductor: \(2.27821\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{650} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 650,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 3 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 13 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
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

−10.33981883717841530698728832250, −9.306420860153218895558185083831, −8.470593426325214029406055436314, −6.90149710443616766187323727244, −6.50957455206104884247861007412, −5.66074294846131677349398169974, −4.55255106907027843064175239055, −3.46602077604760174991666093933, −2.38389183928965643037856764307, 0, 2.38389183928965643037856764307, 3.46602077604760174991666093933, 4.55255106907027843064175239055, 5.66074294846131677349398169974, 6.50957455206104884247861007412, 6.90149710443616766187323727244, 8.470593426325214029406055436314, 9.306420860153218895558185083831, 10.33981883717841530698728832250

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