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Properties

Label 2-245-7.4-c3-0-24
Degree $2$
Conductor $245$
Sign $0.827 - 0.561i$
Analytic cond. $14.4554$
Root an. cond. $3.80203$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.44 + 2.49i)2-s + (1.44 − 2.50i)3-s + (−0.149 + 0.259i)4-s + (2.5 + 4.33i)5-s + 8.32·6-s + 22.1·8-s + (9.32 + 16.1i)9-s + (−7.20 + 12.4i)10-s + (23.2 − 40.2i)11-s + (0.432 + 0.749i)12-s − 31.0·13-s + 14.4·15-s + (33.1 + 57.4i)16-s + (30.9 − 53.5i)17-s + (−26.8 + 46.5i)18-s + (−12.3 − 21.3i)19-s + ⋯
L(s)  = 1  + (0.509 + 0.882i)2-s + (0.278 − 0.481i)3-s + (−0.0187 + 0.0324i)4-s + (0.223 + 0.387i)5-s + 0.566·6-s + 0.980·8-s + (0.345 + 0.598i)9-s + (−0.227 + 0.394i)10-s + (0.637 − 1.10i)11-s + (0.0104 + 0.0180i)12-s − 0.662·13-s + 0.248·15-s + (0.518 + 0.897i)16-s + (0.441 − 0.764i)17-s + (−0.351 + 0.609i)18-s + (−0.148 − 0.257i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.827 - 0.561i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (116, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ 0.827 - 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.00750 + 0.924115i\)
\(L(\frac12)\) \(\approx\) \(3.00750 + 0.924115i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-2.5 - 4.33i)T \)
7 \( 1 \)
good2 \( 1 + (-1.44 - 2.49i)T + (-4 + 6.92i)T^{2} \)
3 \( 1 + (-1.44 + 2.50i)T + (-13.5 - 23.3i)T^{2} \)
11 \( 1 + (-23.2 + 40.2i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 31.0T + 2.19e3T^{2} \)
17 \( 1 + (-30.9 + 53.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (12.3 + 21.3i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-77.4 - 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 200.T + 2.43e4T^{2} \)
31 \( 1 + (64.6 - 111. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-38.9 - 67.5i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 235.T + 6.89e4T^{2} \)
43 \( 1 + 278.T + 7.95e4T^{2} \)
47 \( 1 + (184. + 318. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-84.7 + 146. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (345. - 598. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (348. + 603. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (1.16 - 2.02i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + 866.T + 3.57e5T^{2} \)
73 \( 1 + (376. - 651. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (421. + 729. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + 1.44e3T + 5.71e5T^{2} \)
89 \( 1 + (-719. - 1.24e3i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 23.4T + 9.12e5T^{2} \)
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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

−11.78275806077064831052170068489, −10.81313289658259681258893720084, −9.826009424194885463481657001076, −8.505975337101231090502403750547, −7.40513827784770481231265537398, −6.82200288464306634446574181756, −5.70284322402875247314735300286, −4.71257410508571815740488821272, −3.01658740991364639189301381877, −1.38919112541842551148056643581, 1.42021110477406896594134455966, 2.79798083592171096655343449432, 4.12505268196030168669116366335, 4.71370515904168010316987697149, 6.46304831442203583369641372467, 7.62214085070730231581933381607, 8.920347754842364757874307229533, 9.896182348857481793005411544668, 10.50506241002059595784968876213, 11.83727446594433919651422692358

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