Properties

Label 2-75-1.1-c9-0-13
Degree $2$
Conductor $75$
Sign $-1$
Analytic cond. $38.6276$
Root an. cond. $6.21511$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 18·2-s − 81·3-s − 188·4-s + 1.45e3·6-s − 9.12e3·7-s + 1.26e4·8-s + 6.56e3·9-s + 2.11e4·11-s + 1.52e4·12-s − 3.12e4·13-s + 1.64e5·14-s − 1.30e5·16-s + 2.79e5·17-s − 1.18e5·18-s + 1.44e5·19-s + 7.39e5·21-s − 3.80e5·22-s + 1.76e6·23-s − 1.02e6·24-s + 5.61e5·26-s − 5.31e5·27-s + 1.71e6·28-s + 4.69e6·29-s − 3.69e5·31-s − 4.10e6·32-s − 1.71e6·33-s − 5.02e6·34-s + ⋯
L(s)  = 1  − 0.795·2-s − 0.577·3-s − 0.367·4-s + 0.459·6-s − 1.43·7-s + 1.08·8-s + 1/3·9-s + 0.435·11-s + 0.211·12-s − 0.303·13-s + 1.14·14-s − 0.497·16-s + 0.811·17-s − 0.265·18-s + 0.253·19-s + 0.829·21-s − 0.346·22-s + 1.31·23-s − 0.627·24-s + 0.241·26-s − 0.192·27-s + 0.527·28-s + 1.23·29-s − 0.0717·31-s − 0.691·32-s − 0.251·33-s − 0.645·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $-1$
Analytic conductor: \(38.6276\)
Root analytic conductor: \(6.21511\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 75,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + p^{4} T \)
5 \( 1 \)
good2 \( 1 + 9 p T + p^{9} T^{2} \)
7 \( 1 + 1304 p T + p^{9} T^{2} \)
11 \( 1 - 21132 T + p^{9} T^{2} \)
13 \( 1 + 31214 T + p^{9} T^{2} \)
17 \( 1 - 279342 T + p^{9} T^{2} \)
19 \( 1 - 7580 p T + p^{9} T^{2} \)
23 \( 1 - 1763496 T + p^{9} T^{2} \)
29 \( 1 - 4692510 T + p^{9} T^{2} \)
31 \( 1 + 369088 T + p^{9} T^{2} \)
37 \( 1 + 9347078 T + p^{9} T^{2} \)
41 \( 1 + 7226838 T + p^{9} T^{2} \)
43 \( 1 - 23147476 T + p^{9} T^{2} \)
47 \( 1 + 22971888 T + p^{9} T^{2} \)
53 \( 1 + 78477174 T + p^{9} T^{2} \)
59 \( 1 + 20310660 T + p^{9} T^{2} \)
61 \( 1 + 179339938 T + p^{9} T^{2} \)
67 \( 1 + 274528388 T + p^{9} T^{2} \)
71 \( 1 + 36342648 T + p^{9} T^{2} \)
73 \( 1 - 247089526 T + p^{9} T^{2} \)
79 \( 1 - 191874800 T + p^{9} T^{2} \)
83 \( 1 - 276159276 T + p^{9} T^{2} \)
89 \( 1 + 678997350 T + p^{9} T^{2} \)
97 \( 1 - 567657502 T + p^{9} T^{2} \)
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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

−12.15825359068109263626080881677, −10.65698519943216617918101260072, −9.784018101590613090355135286061, −8.985447999303704123671122878709, −7.46079058412585437526921945885, −6.33950216747201877527550041183, −4.85386783735287665983183739105, −3.28649262451312788450151086465, −1.11917068055141386823081433814, 0, 1.11917068055141386823081433814, 3.28649262451312788450151086465, 4.85386783735287665983183739105, 6.33950216747201877527550041183, 7.46079058412585437526921945885, 8.985447999303704123671122878709, 9.784018101590613090355135286061, 10.65698519943216617918101260072, 12.15825359068109263626080881677

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