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Properties

Label 2-245-35.13-c1-0-15
Degree $2$
Conductor $245$
Sign $-0.298 + 0.954i$
Analytic cond. $1.95633$
Root an. cond. $1.39869$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.36 − 1.36i)2-s + (1.36 − 1.36i)3-s − 1.73i·4-s + (−2 − i)5-s − 3.73i·6-s + (0.366 + 0.366i)8-s − 0.732i·9-s + (−4.09 + 1.36i)10-s + 0.732·11-s + (−2.36 − 2.36i)12-s + (−2 + 2i)13-s + (−4.09 + 1.36i)15-s + 4.46·16-s + (0.732 + 0.732i)17-s + (−0.999 − 0.999i)18-s + 2.73·19-s + ⋯
L(s)  = 1  + (0.965 − 0.965i)2-s + (0.788 − 0.788i)3-s − 0.866i·4-s + (−0.894 − 0.447i)5-s − 1.52i·6-s + (0.129 + 0.129i)8-s − 0.244i·9-s + (−1.29 + 0.431i)10-s + 0.220·11-s + (−0.683 − 0.683i)12-s + (−0.554 + 0.554i)13-s + (−1.05 + 0.352i)15-s + 1.11·16-s + (0.177 + 0.177i)17-s + (−0.235 − 0.235i)18-s + 0.626·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.298 + 0.954i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ -0.298 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31961 - 1.79607i\)
\(L(\frac12)\) \(\approx\) \(1.31961 - 1.79607i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2 + i)T \)
7 \( 1 \)
good2 \( 1 + (-1.36 + 1.36i)T - 2iT^{2} \)
3 \( 1 + (-1.36 + 1.36i)T - 3iT^{2} \)
11 \( 1 - 0.732T + 11T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (-0.732 - 0.732i)T + 17iT^{2} \)
19 \( 1 - 2.73T + 19T^{2} \)
23 \( 1 + (5.09 + 5.09i)T + 23iT^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 - 0.535iT - 31T^{2} \)
37 \( 1 + (3.46 - 3.46i)T - 37iT^{2} \)
41 \( 1 + 0.464iT - 41T^{2} \)
43 \( 1 + (5.83 + 5.83i)T + 43iT^{2} \)
47 \( 1 + (0.464 + 0.464i)T + 47iT^{2} \)
53 \( 1 + (5 + 5i)T + 53iT^{2} \)
59 \( 1 + 2.19T + 59T^{2} \)
61 \( 1 - 8.46iT - 61T^{2} \)
67 \( 1 + (0.830 - 0.830i)T - 67iT^{2} \)
71 \( 1 - 4.73T + 71T^{2} \)
73 \( 1 + (2.53 - 2.53i)T - 73iT^{2} \)
79 \( 1 - 6.73iT - 79T^{2} \)
83 \( 1 + (-3.09 + 3.09i)T - 83iT^{2} \)
89 \( 1 + 16.6T + 89T^{2} \)
97 \( 1 + (-7.92 - 7.92i)T + 97iT^{2} \)
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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

−12.16066449366402388097495320946, −11.29413452317183891234908519403, −10.13075089973878058106460514050, −8.691759881294683334297403684397, −7.935978167230126076750802931832, −6.93716330113541154079966650838, −5.15217499296133679029097094006, −4.10693277585559435231597070057, −2.99675848133780814309802307991, −1.70239968184969128747197272934, 3.17855357053246649828521883377, 3.94561733153180903237663335026, 4.98204515741152713416999143793, 6.26557337736235824925591136867, 7.46632750674137282258092335135, 8.080342089127281898290420245919, 9.490068861282876909307142238980, 10.30126576910575672360891730556, 11.63899989517937222657120943779, 12.52960694546886805302151349741

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