Properties

Label 2-75-1.1-c21-0-10
Degree $2$
Conductor $75$
Sign $1$
Analytic cond. $209.608$
Root an. cond. $14.4778$
Motivic weight $21$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.72e3·2-s + 5.90e4·3-s + 8.88e5·4-s − 1.02e8·6-s − 5.38e8·7-s + 2.08e9·8-s + 3.48e9·9-s − 6.41e10·11-s + 5.24e10·12-s + 1.30e11·13-s + 9.30e11·14-s − 5.47e12·16-s − 8.24e12·17-s − 6.02e12·18-s + 1.34e13·19-s − 3.17e13·21-s + 1.10e14·22-s + 2.33e14·23-s + 1.23e14·24-s − 2.26e14·26-s + 2.05e14·27-s − 4.78e14·28-s − 2.02e15·29-s − 6.86e15·31-s + 5.07e15·32-s − 3.78e15·33-s + 1.42e16·34-s + ⋯
L(s)  = 1  − 1.19·2-s + 0.577·3-s + 0.423·4-s − 0.688·6-s − 0.720·7-s + 0.687·8-s + 1/3·9-s − 0.745·11-s + 0.244·12-s + 0.263·13-s + 0.859·14-s − 1.24·16-s − 0.991·17-s − 0.397·18-s + 0.504·19-s − 0.415·21-s + 0.889·22-s + 1.17·23-s + 0.396·24-s − 0.314·26-s + 0.192·27-s − 0.305·28-s − 0.893·29-s − 1.50·31-s + 0.797·32-s − 0.430·33-s + 1.18·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(75\)    =    \(3 \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(209.608\)
Root analytic conductor: \(14.4778\)
Motivic weight: \(21\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 75,\ (\ :21/2),\ 1)\)

Particular Values

\(L(11)\) \(\approx\) \(0.7187786657\)
\(L(\frac12)\) \(\approx\) \(0.7187786657\)
\(L(\frac{23}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p^{10} T \)
5 \( 1 \)
good2 \( 1 + 27 p^{6} T + p^{21} T^{2} \)
7 \( 1 + 76918544 p T + p^{21} T^{2} \)
11 \( 1 + 64113040188 T + p^{21} T^{2} \)
13 \( 1 - 10075392922 p T + p^{21} T^{2} \)
17 \( 1 + 8242029723618 T + p^{21} T^{2} \)
19 \( 1 - 710110618580 p T + p^{21} T^{2} \)
23 \( 1 - 233184825844776 T + p^{21} T^{2} \)
29 \( 1 + 2024562031123770 T + p^{21} T^{2} \)
31 \( 1 + 6869194988701768 T + p^{21} T^{2} \)
37 \( 1 + 3443998107027638 T + p^{21} T^{2} \)
41 \( 1 + 21842403084625158 T + p^{21} T^{2} \)
43 \( 1 - 71792816814133756 T + p^{21} T^{2} \)
47 \( 1 + 283544719418655648 T + p^{21} T^{2} \)
53 \( 1 - 2172285419049898146 T + p^{21} T^{2} \)
59 \( 1 - 1534831476719068260 T + p^{21} T^{2} \)
61 \( 1 - 4311589520797626062 T + p^{21} T^{2} \)
67 \( 1 + 9243910904037307868 T + p^{21} T^{2} \)
71 \( 1 + 20387361256404760728 T + p^{21} T^{2} \)
73 \( 1 + 16617754439328636074 T + p^{21} T^{2} \)
79 \( 1 - 67940304745507627880 T + p^{21} T^{2} \)
83 \( 1 + 39503732340682314684 T + p^{21} T^{2} \)
89 \( 1 - 41611676186839694490 T + p^{21} T^{2} \)
97 \( 1 + 57181473208903260098 T + p^{21} T^{2} \)
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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

−10.34636485771363608691888853453, −9.343774394065293036927750667166, −8.763023026799325630342744834256, −7.63192211652867575206940536021, −6.83425875551027798180949754532, −5.23877853268689124892791676580, −3.87056402560267868688326422147, −2.67649576341684120148123580297, −1.60371098125008608318554100159, −0.41355514769802869946612605637, 0.41355514769802869946612605637, 1.60371098125008608318554100159, 2.67649576341684120148123580297, 3.87056402560267868688326422147, 5.23877853268689124892791676580, 6.83425875551027798180949754532, 7.63192211652867575206940536021, 8.763023026799325630342744834256, 9.343774394065293036927750667166, 10.34636485771363608691888853453

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