Properties

Label 2-245-1.1-c5-0-7
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  − 8·2-s − 3-s + 32·4-s − 25·5-s + 8·6-s − 242·9-s + 200·10-s − 453·11-s − 32·12-s + 969·13-s + 25·15-s − 1.02e3·16-s − 1.63e3·17-s + 1.93e3·18-s + 1.55e3·19-s − 800·20-s + 3.62e3·22-s − 1.65e3·23-s + 625·25-s − 7.75e3·26-s + 485·27-s − 4.98e3·29-s − 200·30-s − 1.19e3·31-s + 8.19e3·32-s + 453·33-s + 1.30e4·34-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.0641·3-s + 4-s − 0.447·5-s + 0.0907·6-s − 0.995·9-s + 0.632·10-s − 1.12·11-s − 0.0641·12-s + 1.59·13-s + 0.0286·15-s − 16-s − 1.37·17-s + 1.40·18-s + 0.985·19-s − 0.447·20-s + 1.59·22-s − 0.651·23-s + 1/5·25-s − 2.24·26-s + 0.128·27-s − 1.10·29-s − 0.0405·30-s − 0.222·31-s + 1.41·32-s + 0.0724·33-s + 1.94·34-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

\(L(3)\) \(\approx\) \(0.4079287788\)
\(L(\frac12)\) \(\approx\) \(0.4079287788\)
\(L(\frac{7}{2})\) 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 \( 1 + p^{3} T + p^{5} T^{2} \)
3 \( 1 + T + p^{5} T^{2} \)
11 \( 1 + 453 T + p^{5} T^{2} \)
13 \( 1 - 969 T + p^{5} T^{2} \)
17 \( 1 + 1637 T + p^{5} T^{2} \)
19 \( 1 - 1550 T + p^{5} T^{2} \)
23 \( 1 + 1654 T + p^{5} T^{2} \)
29 \( 1 + 4985 T + p^{5} T^{2} \)
31 \( 1 + 1192 T + p^{5} T^{2} \)
37 \( 1 + 11018 T + p^{5} T^{2} \)
41 \( 1 - 1728 T + p^{5} T^{2} \)
43 \( 1 + 10814 T + p^{5} T^{2} \)
47 \( 1 + 26237 T + p^{5} T^{2} \)
53 \( 1 - 25936 T + p^{5} T^{2} \)
59 \( 1 - 4580 T + p^{5} T^{2} \)
61 \( 1 - 12488 T + p^{5} T^{2} \)
67 \( 1 + 15848 T + p^{5} T^{2} \)
71 \( 1 - 51792 T + p^{5} T^{2} \)
73 \( 1 + 4846 T + p^{5} T^{2} \)
79 \( 1 - 62765 T + p^{5} T^{2} \)
83 \( 1 - 23644 T + p^{5} T^{2} \)
89 \( 1 - 147300 T + p^{5} T^{2} \)
97 \( 1 - 8343 T + p^{5} T^{2} \)
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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

−11.09054695425803143104270803503, −10.28020073412302408694023285859, −9.041986878948939935928688889782, −8.439833256553057780905651756435, −7.67358664499193293046329060050, −6.47306982880801984971813779089, −5.17476642122757749835724493930, −3.51970268882922076779846921782, −2.00020368014319854163554978379, −0.45177147819378609656417548347, 0.45177147819378609656417548347, 2.00020368014319854163554978379, 3.51970268882922076779846921782, 5.17476642122757749835724493930, 6.47306982880801984971813779089, 7.67358664499193293046329060050, 8.439833256553057780905651756435, 9.041986878948939935928688889782, 10.28020073412302408694023285859, 11.09054695425803143104270803503

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