Properties

Label 2-650-1.1-c5-0-85
Degree $2$
Conductor $650$
Sign $-1$
Analytic cond. $104.249$
Root an. cond. $10.2102$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 170·7-s + 64·8-s − 243·9-s − 250·11-s + 169·13-s + 680·14-s + 256·16-s − 1.06e3·17-s − 972·18-s − 78·19-s − 1.00e3·22-s − 1.57e3·23-s + 676·26-s + 2.72e3·28-s + 2.57e3·29-s − 8.65e3·31-s + 1.02e3·32-s − 4.24e3·34-s − 3.88e3·36-s − 1.09e4·37-s − 312·38-s + 1.05e3·41-s + 5.90e3·43-s − 4.00e3·44-s − 6.30e3·46-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 1.31·7-s + 0.353·8-s − 9-s − 0.622·11-s + 0.277·13-s + 0.927·14-s + 1/4·16-s − 0.891·17-s − 0.707·18-s − 0.0495·19-s − 0.440·22-s − 0.621·23-s + 0.196·26-s + 0.655·28-s + 0.569·29-s − 1.61·31-s + 0.176·32-s − 0.630·34-s − 1/2·36-s − 1.31·37-s − 0.0350·38-s + 0.0975·41-s + 0.486·43-s − 0.311·44-s − 0.439·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(650\)    =    \(2 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(104.249\)
Root analytic conductor: \(10.2102\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 650,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
5 \( 1 \)
13 \( 1 - p^{2} T \)
good3 \( 1 + p^{5} T^{2} \)
7 \( 1 - 170 T + p^{5} T^{2} \)
11 \( 1 + 250 T + p^{5} T^{2} \)
17 \( 1 + 1062 T + p^{5} T^{2} \)
19 \( 1 + 78 T + p^{5} T^{2} \)
23 \( 1 + 1576 T + p^{5} T^{2} \)
29 \( 1 - 2578 T + p^{5} T^{2} \)
31 \( 1 + 8654 T + p^{5} T^{2} \)
37 \( 1 + 10986 T + p^{5} T^{2} \)
41 \( 1 - 1050 T + p^{5} T^{2} \)
43 \( 1 - 5900 T + p^{5} T^{2} \)
47 \( 1 - 5962 T + p^{5} T^{2} \)
53 \( 1 + 29046 T + p^{5} T^{2} \)
59 \( 1 + 13922 T + p^{5} T^{2} \)
61 \( 1 + 32882 T + p^{5} T^{2} \)
67 \( 1 - 69566 T + p^{5} T^{2} \)
71 \( 1 + 50542 T + p^{5} T^{2} \)
73 \( 1 - 46750 T + p^{5} T^{2} \)
79 \( 1 + 19348 T + p^{5} T^{2} \)
83 \( 1 - 87438 T + p^{5} T^{2} \)
89 \( 1 - 94170 T + p^{5} T^{2} \)
97 \( 1 + 182786 T + p^{5} T^{2} \)
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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

−9.202556394200829785741912302863, −8.319695105908089468027003881780, −7.66853289368567459908782992876, −6.49668158697562651722023255117, −5.49015587431006008423933728563, −4.87402518910510669989378984395, −3.78391655139048249736452426617, −2.56364027077117633551166565885, −1.64769560152536882980931007954, 0, 1.64769560152536882980931007954, 2.56364027077117633551166565885, 3.78391655139048249736452426617, 4.87402518910510669989378984395, 5.49015587431006008423933728563, 6.49668158697562651722023255117, 7.66853289368567459908782992876, 8.319695105908089468027003881780, 9.202556394200829785741912302863

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