Properties

Label 2-245-1.1-c5-0-12
Degree $2$
Conductor $245$
Sign $1$
Analytic cond. $39.2940$
Root an. cond. $6.26849$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 4·3-s − 28·4-s − 25·5-s + 8·6-s − 120·8-s − 227·9-s − 50·10-s − 148·11-s − 112·12-s − 286·13-s − 100·15-s + 656·16-s + 1.67e3·17-s − 454·18-s − 1.06e3·19-s + 700·20-s − 296·22-s + 2.97e3·23-s − 480·24-s + 625·25-s − 572·26-s − 1.88e3·27-s − 3.41e3·29-s − 200·30-s + 2.44e3·31-s + 5.15e3·32-s + ⋯
L(s)  = 1  + 0.353·2-s + 0.256·3-s − 7/8·4-s − 0.447·5-s + 0.0907·6-s − 0.662·8-s − 0.934·9-s − 0.158·10-s − 0.368·11-s − 0.224·12-s − 0.469·13-s − 0.114·15-s + 0.640·16-s + 1.40·17-s − 0.330·18-s − 0.673·19-s + 0.391·20-s − 0.130·22-s + 1.17·23-s − 0.170·24-s + 1/5·25-s − 0.165·26-s − 0.496·27-s − 0.752·29-s − 0.0405·30-s + 0.457·31-s + 0.889·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(39.2940\)
Root analytic conductor: \(6.26849\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: $\chi_{245} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(1.362191020\)
\(L(\frac12)\) \(\approx\) \(1.362191020\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + p^{2} T \)
7 \( 1 \)
good2 \( 1 - p T + p^{5} T^{2} \)
3 \( 1 - 4 T + p^{5} T^{2} \)
11 \( 1 + 148 T + p^{5} T^{2} \)
13 \( 1 + 22 p T + p^{5} T^{2} \)
17 \( 1 - 1678 T + p^{5} T^{2} \)
19 \( 1 + 1060 T + p^{5} T^{2} \)
23 \( 1 - 2976 T + p^{5} T^{2} \)
29 \( 1 + 3410 T + p^{5} T^{2} \)
31 \( 1 - 2448 T + p^{5} T^{2} \)
37 \( 1 - 182 T + p^{5} T^{2} \)
41 \( 1 - 9398 T + p^{5} T^{2} \)
43 \( 1 + 1244 T + p^{5} T^{2} \)
47 \( 1 - 12088 T + p^{5} T^{2} \)
53 \( 1 - 23846 T + p^{5} T^{2} \)
59 \( 1 - 20020 T + p^{5} T^{2} \)
61 \( 1 + 32302 T + p^{5} T^{2} \)
67 \( 1 - 60972 T + p^{5} T^{2} \)
71 \( 1 + 32648 T + p^{5} T^{2} \)
73 \( 1 - 38774 T + p^{5} T^{2} \)
79 \( 1 + 33360 T + p^{5} T^{2} \)
83 \( 1 + 16716 T + p^{5} T^{2} \)
89 \( 1 + 101370 T + p^{5} T^{2} \)
97 \( 1 - 119038 T + p^{5} T^{2} \)
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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

−11.36486056211701427070013667621, −10.20391635548427364711426688711, −9.163127495753059572419585194562, −8.365629225637249283591375542418, −7.44717982173555021370698562607, −5.85071453229782270722461301097, −5.00267729164143909410617395082, −3.76579732720822462358053902204, −2.75311277071952905253925310886, −0.63450262739949142060609660852, 0.63450262739949142060609660852, 2.75311277071952905253925310886, 3.76579732720822462358053902204, 5.00267729164143909410617395082, 5.85071453229782270722461301097, 7.44717982173555021370698562607, 8.365629225637249283591375542418, 9.163127495753059572419585194562, 10.20391635548427364711426688711, 11.36486056211701427070013667621

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