Properties

Label 2-7350-1.1-c1-0-47
Degree $2$
Conductor $7350$
Sign $1$
Analytic cond. $58.6900$
Root an. cond. $7.66094$
Motivic weight $1$
Arithmetic yes
Rational no
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 + 1.67·11-s + 12-s + 4.48·13-s + 16-s + 7.61·17-s − 18-s − 4.48·19-s − 1.67·22-s + 0.482·23-s − 24-s − 4.48·26-s + 27-s + 1.19·29-s + 3.48·31-s − 32-s + 1.67·33-s − 7.61·34-s + 36-s + 0.482·37-s + 4.48·38-s + 4.48·39-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 + 0.505·11-s + 0.288·12-s + 1.24·13-s + 0.250·16-s + 1.84·17-s − 0.235·18-s − 1.02·19-s − 0.357·22-s + 0.100·23-s − 0.204·24-s − 0.879·26-s + 0.192·27-s + 0.221·29-s + 0.625·31-s − 0.176·32-s + 0.291·33-s − 1.30·34-s + 0.166·36-s + 0.0793·37-s + 0.727·38-s + 0.717·39-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 7350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.262943140\)
\(L(\frac12)\) \(\approx\) \(2.262943140\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.67T + 11T^{2} \)
13 \( 1 - 4.48T + 13T^{2} \)
17 \( 1 - 7.61T + 17T^{2} \)
19 \( 1 + 4.48T + 19T^{2} \)
23 \( 1 - 0.482T + 23T^{2} \)
29 \( 1 - 1.19T + 29T^{2} \)
31 \( 1 - 3.48T + 31T^{2} \)
37 \( 1 - 0.482T + 37T^{2} \)
41 \( 1 - 12.0T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 11.4T + 47T^{2} \)
53 \( 1 + 9.57T + 53T^{2} \)
59 \( 1 + 5.77T + 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 - 3.35T + 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4T + 73T^{2} \)
79 \( 1 - 4.51T + 79T^{2} \)
83 \( 1 - 1.87T + 83T^{2} \)
89 \( 1 - 3.35T + 89T^{2} \)
97 \( 1 + 17.7T + 97T^{2} \)
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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

−8.109849765988647667987016497947, −7.44973852726647826342426261472, −6.46491428271005997429727206574, −6.14303483901196356682511959974, −5.14237725697498262382222389656, −4.08234734750938775145370988313, −3.46776235900505420172486945929, −2.66677727560334043619551224712, −1.60123625883473819296971931963, −0.891879589469407284624940548546, 0.891879589469407284624940548546, 1.60123625883473819296971931963, 2.66677727560334043619551224712, 3.46776235900505420172486945929, 4.08234734750938775145370988313, 5.14237725697498262382222389656, 6.14303483901196356682511959974, 6.46491428271005997429727206574, 7.44973852726647826342426261472, 8.109849765988647667987016497947

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