Properties

Label 2-50-1.1-c5-0-6
Degree $2$
Conductor $50$
Sign $-1$
Analytic cond. $8.01919$
Root an. cond. $2.83181$
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 − 14·3-s + 16·4-s − 56·6-s − 158·7-s + 64·8-s − 47·9-s − 148·11-s − 224·12-s − 684·13-s − 632·14-s + 256·16-s − 2.04e3·17-s − 188·18-s + 2.22e3·19-s + 2.21e3·21-s − 592·22-s + 1.24e3·23-s − 896·24-s − 2.73e3·26-s + 4.06e3·27-s − 2.52e3·28-s − 270·29-s − 2.04e3·31-s + 1.02e3·32-s + 2.07e3·33-s − 8.19e3·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.898·3-s + 1/2·4-s − 0.635·6-s − 1.21·7-s + 0.353·8-s − 0.193·9-s − 0.368·11-s − 0.449·12-s − 1.12·13-s − 0.861·14-s + 1/4·16-s − 1.71·17-s − 0.136·18-s + 1.41·19-s + 1.09·21-s − 0.260·22-s + 0.491·23-s − 0.317·24-s − 0.793·26-s + 1.07·27-s − 0.609·28-s − 0.0596·29-s − 0.382·31-s + 0.176·32-s + 0.331·33-s − 1.21·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50\)    =    \(2 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(8.01919\)
Root analytic conductor: \(2.83181\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 50,\ (\ :5/2),\ -1)\)

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 \( 1 \)
good3 \( 1 + 14 T + p^{5} T^{2} \)
7 \( 1 + 158 T + p^{5} T^{2} \)
11 \( 1 + 148 T + p^{5} T^{2} \)
13 \( 1 + 684 T + p^{5} T^{2} \)
17 \( 1 + 2048 T + p^{5} T^{2} \)
19 \( 1 - 2220 T + p^{5} T^{2} \)
23 \( 1 - 1246 T + p^{5} T^{2} \)
29 \( 1 + 270 T + p^{5} T^{2} \)
31 \( 1 + 2048 T + p^{5} T^{2} \)
37 \( 1 - 4372 T + p^{5} T^{2} \)
41 \( 1 + 2398 T + p^{5} T^{2} \)
43 \( 1 + 2294 T + p^{5} T^{2} \)
47 \( 1 - 10682 T + p^{5} T^{2} \)
53 \( 1 + 2964 T + p^{5} T^{2} \)
59 \( 1 + 39740 T + p^{5} T^{2} \)
61 \( 1 + 42298 T + p^{5} T^{2} \)
67 \( 1 + 32098 T + p^{5} T^{2} \)
71 \( 1 + 4248 T + p^{5} T^{2} \)
73 \( 1 + 30104 T + p^{5} T^{2} \)
79 \( 1 - 35280 T + p^{5} T^{2} \)
83 \( 1 - 27826 T + p^{5} T^{2} \)
89 \( 1 + 85210 T + p^{5} T^{2} \)
97 \( 1 - 97232 T + p^{5} T^{2} \)
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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

−13.76687182289615273782577049366, −12.76116946830580822704485861269, −11.80402499298806288614731212739, −10.69679989356991589984226517485, −9.347621679306487193878717081243, −7.19737613179483278704107044851, −6.08295761060109529882533052878, −4.83679840622951713964954782483, −2.88836097607258925743486871536, 0, 2.88836097607258925743486871536, 4.83679840622951713964954782483, 6.08295761060109529882533052878, 7.19737613179483278704107044851, 9.347621679306487193878717081243, 10.69679989356991589984226517485, 11.80402499298806288614731212739, 12.76116946830580822704485861269, 13.76687182289615273782577049366

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