Properties

Label 2-7350-1.1-c1-0-82
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 − 2·11-s − 12-s + 2·13-s + 16-s + 8·17-s − 18-s + 2·19-s + 2·22-s + 24-s − 2·26-s − 27-s − 6·29-s − 6·31-s − 32-s + 2·33-s − 8·34-s + 36-s − 8·37-s − 2·38-s − 2·39-s − 6·41-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 − 0.603·11-s − 0.288·12-s + 0.554·13-s + 1/4·16-s + 1.94·17-s − 0.235·18-s + 0.458·19-s + 0.426·22-s + 0.204·24-s − 0.392·26-s − 0.192·27-s − 1.11·29-s − 1.07·31-s − 0.176·32-s + 0.348·33-s − 1.37·34-s + 1/6·36-s − 1.31·37-s − 0.324·38-s − 0.320·39-s − 0.937·41-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: $-1$
Analytic conductor: \(58.6900\)
Root analytic conductor: \(7.66094\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{7350} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 8 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 14 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 - 16 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−7.49108116922206628652924050507, −7.11037158741530121044394188469, −6.11259524929670224967255950549, −5.50595512403795059422829163865, −5.04577393647344934936398131026, −3.68808758256329429553333999313, −3.24135258361572790991903833473, −1.95020484992179913191857582097, −1.16381327402761241330659221963, 0, 1.16381327402761241330659221963, 1.95020484992179913191857582097, 3.24135258361572790991903833473, 3.68808758256329429553333999313, 5.04577393647344934936398131026, 5.50595512403795059422829163865, 6.11259524929670224967255950549, 7.11037158741530121044394188469, 7.49108116922206628652924050507

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