Properties

Label 2-25215-1.1-c1-0-7
Degree $2$
Conductor $25215$
Sign $-1$
Analytic cond. $201.342$
Root an. cond. $14.1895$
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 + 5-s − 6-s + 3·8-s + 9-s − 10-s + 4·11-s − 12-s + 2·13-s + 15-s − 16-s − 2·17-s − 18-s − 4·19-s − 20-s − 4·22-s + 3·24-s + 25-s − 2·26-s + 27-s + 2·29-s − 30-s − 5·32-s + 4·33-s + 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 1.20·11-s − 0.288·12-s + 0.554·13-s + 0.258·15-s − 1/4·16-s − 0.485·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s − 0.852·22-s + 0.612·24-s + 1/5·25-s − 0.392·26-s + 0.192·27-s + 0.371·29-s − 0.182·30-s − 0.883·32-s + 0.696·33-s + 0.342·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25215 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25215\)    =    \(3 \cdot 5 \cdot 41^{2}\)
Sign: $-1$
Analytic conductor: \(201.342\)
Root analytic conductor: \(14.1895\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{25215} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25215,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
5 \( 1 - T \)
41 \( 1 \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−15.68164394642440, −14.89848863551871, −14.61734301190701, −13.88969842821989, −13.61464233004950, −13.12379677875317, −12.41166118757646, −11.93336562795837, −10.97713790470647, −10.69010247357684, −10.02449228050021, −9.407760481585933, −9.071523639302205, −8.545105119038792, −8.161497195449445, −7.377595665832176, −6.615087864771428, −6.356592160396822, −5.310900564854340, −4.702272880546634, −3.960654254335747, −3.581212074483096, −2.500626019742822, −1.686254358439200, −1.186747828672650, 0, 1.186747828672650, 1.686254358439200, 2.500626019742822, 3.581212074483096, 3.960654254335747, 4.702272880546634, 5.310900564854340, 6.356592160396822, 6.615087864771428, 7.377595665832176, 8.161497195449445, 8.545105119038792, 9.071523639302205, 9.407760481585933, 10.02449228050021, 10.69010247357684, 10.97713790470647, 11.93336562795837, 12.41166118757646, 13.12379677875317, 13.61464233004950, 13.88969842821989, 14.61734301190701, 14.89848863551871, 15.68164394642440

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