Properties

Label 2-245-1.1-c1-0-5
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
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  − 0.414·2-s + 2.41·3-s − 1.82·4-s + 5-s − 0.999·6-s + 1.58·8-s + 2.82·9-s − 0.414·10-s + 4.82·11-s − 4.41·12-s + 0.828·13-s + 2.41·15-s + 3·16-s − 0.828·17-s − 1.17·18-s − 2.82·19-s − 1.82·20-s − 1.99·22-s − 2.41·23-s + 3.82·24-s + 25-s − 0.343·26-s − 0.414·27-s − 29-s − 0.999·30-s − 6·31-s − 4.41·32-s + ⋯
L(s)  = 1  − 0.292·2-s + 1.39·3-s − 0.914·4-s + 0.447·5-s − 0.408·6-s + 0.560·8-s + 0.942·9-s − 0.130·10-s + 1.45·11-s − 1.27·12-s + 0.229·13-s + 0.623·15-s + 0.750·16-s − 0.200·17-s − 0.276·18-s − 0.648·19-s − 0.408·20-s − 0.426·22-s − 0.503·23-s + 0.781·24-s + 0.200·25-s − 0.0672·26-s − 0.0797·27-s − 0.185·29-s − 0.182·30-s − 1.07·31-s − 0.780·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.513838173\)
\(L(\frac12)\) \(\approx\) \(1.513838173\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 - T \)
7 \( 1 \)
good2 \( 1 + 0.414T + 2T^{2} \)
3 \( 1 - 2.41T + 3T^{2} \)
11 \( 1 - 4.82T + 11T^{2} \)
13 \( 1 - 0.828T + 13T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 2.41T + 23T^{2} \)
29 \( 1 + T + 29T^{2} \)
31 \( 1 + 6T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 2.17T + 41T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 - 2T + 47T^{2} \)
53 \( 1 + 6.82T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 11.4T + 61T^{2} \)
67 \( 1 - 12.4T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 - 4.82T + 73T^{2} \)
79 \( 1 - 9.17T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 2.65T + 89T^{2} \)
97 \( 1 - 0.343T + 97T^{2} \)
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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

−12.37657961323749241891053169558, −10.90182647674348249924956185325, −9.680104268635009568238281529487, −9.128517205586172767917323920038, −8.519614347591273236320156022142, −7.46214629350117722925744398268, −6.06758895217091148698720327366, −4.41142485206502083151230136290, −3.47719629566148823841163821424, −1.75970914726115654218644943864, 1.75970914726115654218644943864, 3.47719629566148823841163821424, 4.41142485206502083151230136290, 6.06758895217091148698720327366, 7.46214629350117722925744398268, 8.519614347591273236320156022142, 9.128517205586172767917323920038, 9.680104268635009568238281529487, 10.90182647674348249924956185325, 12.37657961323749241891053169558

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