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 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 12-s − 3·13-s + 16-s − 3·17-s + 18-s + 2·19-s − 3·23-s − 24-s − 3·26-s − 27-s + 7·29-s − 31-s + 32-s − 3·34-s + 36-s − 4·37-s + 2·38-s + 3·39-s + 5·41-s + 3·43-s − 3·46-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 − 0.288·12-s − 0.832·13-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.458·19-s − 0.625·23-s − 0.204·24-s − 0.588·26-s − 0.192·27-s + 1.29·29-s − 0.179·31-s + 0.176·32-s − 0.514·34-s + 1/6·36-s − 0.657·37-s + 0.324·38-s + 0.480·39-s + 0.780·41-s + 0.457·43-s − 0.442·46-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  =  \(1\)
Selberg data  =  \((2,\ 7350,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   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 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + T + p T^{2} \)
59 \( 1 + 13 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−7.43920378625796822139938495446, −6.66819891423494213680857593990, −6.16269998210514138336943444658, −5.37080641263943816473368232150, −4.71381317804770716097375847995, −4.19147733834174690191710068668, −3.16075665281411124156791371698, −2.38498630059023531200008465133, −1.37402158068312105576221456658, 0, 1.37402158068312105576221456658, 2.38498630059023531200008465133, 3.16075665281411124156791371698, 4.19147733834174690191710068668, 4.71381317804770716097375847995, 5.37080641263943816473368232150, 6.16269998210514138336943444658, 6.66819891423494213680857593990, 7.43920378625796822139938495446

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