Properties

Label 2-9295-1.1-c1-0-100
Degree $2$
Conductor $9295$
Sign $1$
Analytic cond. $74.2209$
Root an. cond. $8.61515$
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  − 2.41·2-s − 2.82·3-s + 3.82·4-s + 5-s + 6.82·6-s + 2·7-s − 4.41·8-s + 5.00·9-s − 2.41·10-s − 11-s − 10.8·12-s − 4.82·14-s − 2.82·15-s + 2.99·16-s + 6.82·17-s − 12.0·18-s + 3.82·20-s − 5.65·21-s + 2.41·22-s − 2.82·23-s + 12.4·24-s + 25-s − 5.65·27-s + 7.65·28-s − 3.65·29-s + 6.82·30-s + 1.58·32-s + ⋯
L(s)  = 1  − 1.70·2-s − 1.63·3-s + 1.91·4-s + 0.447·5-s + 2.78·6-s + 0.755·7-s − 1.56·8-s + 1.66·9-s − 0.763·10-s − 0.301·11-s − 3.12·12-s − 1.29·14-s − 0.730·15-s + 0.749·16-s + 1.65·17-s − 2.84·18-s + 0.856·20-s − 1.23·21-s + 0.514·22-s − 0.589·23-s + 2.54·24-s + 0.200·25-s − 1.08·27-s + 1.44·28-s − 0.679·29-s + 1.24·30-s + 0.280·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9295 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9295\)    =    \(5 \cdot 11 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(74.2209\)
Root analytic conductor: \(8.61515\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{9295} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9295,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5028924419\)
\(L(\frac12)\) \(\approx\) \(0.5028924419\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
11 \( 1 + T \)
13 \( 1 \)
good2 \( 1 + 2.41T + 2T^{2} \)
3 \( 1 + 2.82T + 3T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
17 \( 1 - 6.82T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 7.65T + 37T^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 1.65T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 - 1.17T + 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−7.62926153762886216492375361216, −7.30006999382417069963732032153, −6.30475046427185450448657535679, −5.92897779151536536843196476874, −5.19377388367535862884396994286, −4.53401267986844259817562140131, −3.22219061005423894264054353909, −1.96980343936148011121971291438, −1.35502173047735544451234590375, −0.53629556289382694327575866400, 0.53629556289382694327575866400, 1.35502173047735544451234590375, 1.96980343936148011121971291438, 3.22219061005423894264054353909, 4.53401267986844259817562140131, 5.19377388367535862884396994286, 5.92897779151536536843196476874, 6.30475046427185450448657535679, 7.30006999382417069963732032153, 7.62926153762886216492375361216

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