Properties

Degree 2
Conductor $ 2 \cdot 3 \cdot 5^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s + 3.41·11-s + 12-s + 3·13-s + 16-s + 5.82·17-s − 18-s + 1.41·19-s − 3.41·22-s − 0.414·23-s − 24-s − 3·26-s + 27-s + 29-s − 2.41·31-s − 32-s + 3.41·33-s − 5.82·34-s + 36-s + 7.07·37-s − 1.41·38-s + 3·39-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.408·6-s − 0.353·8-s + 0.333·9-s + 1.02·11-s + 0.288·12-s + 0.832·13-s + 0.250·16-s + 1.41·17-s − 0.235·18-s + 0.324·19-s − 0.727·22-s − 0.0863·23-s − 0.204·24-s − 0.588·26-s + 0.192·27-s + 0.185·29-s − 0.433·31-s − 0.176·32-s + 0.594·33-s − 0.999·34-s + 0.166·36-s + 1.16·37-s − 0.229·38-s + 0.480·39-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

\( d \)  =  \(2\)
\( N \)  =  \(7350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(1\)
character  :  $\chi_{7350} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ 1)\)
\(L(1)\)  \(\approx\)  \(2.347745989\)
\(L(\frac12)\)  \(\approx\)  \(2.347745989\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 3.41T + 11T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 - 5.82T + 17T^{2} \)
19 \( 1 - 1.41T + 19T^{2} \)
23 \( 1 + 0.414T + 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + 2.41T + 31T^{2} \)
37 \( 1 - 7.07T + 37T^{2} \)
41 \( 1 + 5.82T + 41T^{2} \)
43 \( 1 - 1.58T + 43T^{2} \)
47 \( 1 + 1.65T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 - 7.24T + 59T^{2} \)
61 \( 1 + 0.171T + 61T^{2} \)
67 \( 1 + 15.3T + 67T^{2} \)
71 \( 1 + 9.31T + 71T^{2} \)
73 \( 1 - 15.0T + 73T^{2} \)
79 \( 1 - 1.07T + 79T^{2} \)
83 \( 1 + 0.0710T + 83T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 + 0.343T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.897701697077876066231832078797, −7.44609286087223495405952354758, −6.61747002678519463358507992263, −6.00354302515484419868841885436, −5.19454460082498916006587547631, −4.02474321001335692229011899624, −3.51982733141073725737441165224, −2.65931115796620108641636662602, −1.57346745670362878041009768221, −0.922714190537144890624526445936, 0.922714190537144890624526445936, 1.57346745670362878041009768221, 2.65931115796620108641636662602, 3.51982733141073725737441165224, 4.02474321001335692229011899624, 5.19454460082498916006587547631, 6.00354302515484419868841885436, 6.61747002678519463358507992263, 7.44609286087223495405952354758, 7.897701697077876066231832078797

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