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Properties

Label 2-245-35.9-c1-0-2
Degree $2$
Conductor $245$
Sign $0.669 - 0.742i$
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  + (−1.73 − i)2-s + (0.866 − 0.5i)3-s + (0.999 + 1.73i)4-s + (−1.86 + 1.23i)5-s − 1.99·6-s + (−1 + 1.73i)9-s + (4.46 − 0.267i)10-s + (1.5 + 2.59i)11-s + (1.73 + i)12-s + i·13-s + (−1 + 2i)15-s + (1.99 − 3.46i)16-s + (−6.06 + 3.5i)17-s + (3.46 − 2i)18-s + (−3.99 − 2.00i)20-s + ⋯
L(s)  = 1  + (−1.22 − 0.707i)2-s + (0.499 − 0.288i)3-s + (0.499 + 0.866i)4-s + (−0.834 + 0.550i)5-s − 0.816·6-s + (−0.333 + 0.577i)9-s + (1.41 − 0.0847i)10-s + (0.452 + 0.783i)11-s + (0.499 + 0.288i)12-s + 0.277i·13-s + (−0.258 + 0.516i)15-s + (0.499 − 0.866i)16-s + (−1.47 + 0.848i)17-s + (0.816 − 0.471i)18-s + (−0.894 − 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $0.669 - 0.742i$
Analytic conductor: \(1.95633\)
Root analytic conductor: \(1.39869\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :1/2),\ 0.669 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.469165 + 0.208686i\)
\(L(\frac12)\) \(\approx\) \(0.469165 + 0.208686i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 \)
good2 \( 1 + (1.73 + i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.866 + 0.5i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + (6.06 - 3.5i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-5.19 - 3i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 5T + 29T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.73 + i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + (2.59 + 1.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.19 - 3i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 + 8.66i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 + (-5.19 + 3i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7iT - 97T^{2} \)
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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

−11.83056767856817851768799002772, −11.08338144563956736492964780849, −10.42462100214034694635133707677, −9.192768065346644519118327687734, −8.487339097378220730575630121642, −7.62552147484960112274802610856, −6.69308163530968287773729374349, −4.65238035845071197192356430004, −3.06006601498574189296605146131, −1.84734726872874795666210557838, 0.58320189228333347847663258188, 3.24498144029227832381793016210, 4.57496397272411570502517374626, 6.29517662540209609848793514974, 7.22295634002129259118925414961, 8.476293160147693974548809700295, 8.736204522796318873821824377690, 9.549262249037662783242970497672, 10.84639790691903052626172355071, 11.74578875013566843128865157331

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