Properties

Label 2-7350-1.1-c1-0-104
Degree $2$
Conductor $7350$
Sign $-1$
Analytic cond. $58.6900$
Root an. cond. $7.66094$
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 − 6-s + 8-s + 9-s − 4·11-s − 12-s + 4·13-s + 16-s + 3·17-s + 18-s − 6·19-s − 4·22-s − 7·23-s − 24-s + 4·26-s − 27-s + 4·29-s + 5·31-s + 32-s + 4·33-s + 3·34-s + 36-s − 2·37-s − 6·38-s − 4·39-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 1.20·11-s − 0.288·12-s + 1.10·13-s + 1/4·16-s + 0.727·17-s + 0.235·18-s − 1.37·19-s − 0.852·22-s − 1.45·23-s − 0.204·24-s + 0.784·26-s − 0.192·27-s + 0.742·29-s + 0.898·31-s + 0.176·32-s + 0.696·33-s + 0.514·34-s + 1/6·36-s − 0.328·37-s − 0.973·38-s − 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(58.6900\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 17 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−7.47415218255942471976438176842, −6.60220618397810332038818813541, −6.04773962213833599570492747308, −5.54114747838435288152941825677, −4.70321741421989721958914979215, −4.09438452877911047401406976138, −3.24137434564808265370087117479, −2.34718675546875141238127534133, −1.38160626974763524931763379696, 0, 1.38160626974763524931763379696, 2.34718675546875141238127534133, 3.24137434564808265370087117479, 4.09438452877911047401406976138, 4.70321741421989721958914979215, 5.54114747838435288152941825677, 6.04773962213833599570492747308, 6.60220618397810332038818813541, 7.47415218255942471976438176842

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