Properties

Degree $2$
Conductor $7350$
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 − 5·11-s − 12-s − 13-s + 16-s + 2·17-s + 18-s + 7·19-s − 5·22-s − 3·23-s − 24-s − 26-s − 27-s − 6·31-s + 32-s + 5·33-s + 2·34-s + 36-s + 5·37-s + 7·38-s + 39-s − 9·41-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 − 1.50·11-s − 0.288·12-s − 0.277·13-s + 1/4·16-s + 0.485·17-s + 0.235·18-s + 1.60·19-s − 1.06·22-s − 0.625·23-s − 0.204·24-s − 0.196·26-s − 0.192·27-s − 1.07·31-s + 0.176·32-s + 0.870·33-s + 0.342·34-s + 1/6·36-s + 0.821·37-s + 1.13·38-s + 0.160·39-s − 1.40·41-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: $-1$
Motivic weight: \(1\)
Character: $\chi_{7350} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + 5 T + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 5 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 - 10 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−17.36478478843874, −16.58008617461019, −16.21666362410249, −15.68325247455164, −15.09409118705097, −14.49062232558015, −13.64787115743555, −13.36460744555032, −12.68056386751904, −11.97236692198327, −11.68674086222261, −10.84573969181863, −10.25523106506305, −9.847114137912067, −8.902186923145063, −7.859167238819907, −7.557459065099889, −6.832930874244575, −5.900228239496329, −5.332446408451932, −5.015220107748769, −3.990258197809554, −3.187962406510003, −2.429873250944025, −1.343342065644002, 0, 1.343342065644002, 2.429873250944025, 3.187962406510003, 3.990258197809554, 5.015220107748769, 5.332446408451932, 5.900228239496329, 6.832930874244575, 7.557459065099889, 7.859167238819907, 8.902186923145063, 9.847114137912067, 10.25523106506305, 10.84573969181863, 11.68674086222261, 11.97236692198327, 12.68056386751904, 13.36460744555032, 13.64787115743555, 14.49062232558015, 15.09409118705097, 15.68325247455164, 16.21666362410249, 16.58008617461019, 17.36478478843874

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