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Properties

Label 2-245-7.2-c3-0-10
Degree $2$
Conductor $245$
Sign $-0.991 - 0.126i$
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  + (−2.70 + 4.68i)2-s + (2.32 + 4.03i)3-s + (−10.6 − 18.4i)4-s + (2.5 − 4.33i)5-s − 25.2·6-s + 72.0·8-s + (2.65 − 4.60i)9-s + (13.5 + 23.4i)10-s + (26.1 + 45.2i)11-s + (49.6 − 85.9i)12-s + 30.6·13-s + 23.2·15-s + (−109. + 190. i)16-s + (−18.6 − 32.2i)17-s + (14.3 + 24.9i)18-s + (−40.1 + 69.4i)19-s + ⋯
L(s)  = 1  + (−0.957 + 1.65i)2-s + (0.448 + 0.776i)3-s + (−1.33 − 2.30i)4-s + (0.223 − 0.387i)5-s − 1.71·6-s + 3.18·8-s + (0.0984 − 0.170i)9-s + (0.428 + 0.741i)10-s + (0.716 + 1.24i)11-s + (1.19 − 2.06i)12-s + 0.654·13-s + 0.400·15-s + (−1.71 + 2.97i)16-s + (−0.265 − 0.459i)17-s + (0.188 + 0.326i)18-s + (−0.484 + 0.838i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(14.4554\)
Root analytic conductor: \(3.80203\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{245} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.0768063 + 1.21030i\)
\(L(\frac12)\) \(\approx\) \(0.0768063 + 1.21030i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.70 - 4.68i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (-2.32 - 4.03i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-26.1 - 45.2i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 30.6T + 2.19e3T^{2} \)
17 \( 1 + (18.6 + 32.2i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (40.1 - 69.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (12.9 - 22.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 20.9T + 2.43e4T^{2} \)
31 \( 1 + (-157. - 272. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (98.5 - 170. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 11.3T + 6.89e4T^{2} \)
43 \( 1 + 33.8T + 7.95e4T^{2} \)
47 \( 1 + (-180. + 313. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (76.5 + 132. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-308 - 533. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (7.63 - 13.2i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-83.2 - 144. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 952T + 3.57e5T^{2} \)
73 \( 1 + (-74.2 - 128. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (428. - 742. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 660.T + 5.71e5T^{2} \)
89 \( 1 + (-22.8 + 39.6i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.68e3T + 9.12e5T^{2} \)
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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

−12.13409829822778314805687672578, −10.34973008468428770138641011581, −9.875756218150014514505269359943, −8.962368219011550360611478043471, −8.444413408884437876337603695731, −7.12327141954820329546198569252, −6.35141312024303987606372214028, −5.04267602319919802724494576130, −4.11933555023545107173000631738, −1.33654565461070132021952488522, 0.74199361904856778040632325181, 1.96005136476061601856550669498, 2.98913705592743077068522950212, 4.17424417507332596701987388713, 6.38344228501129898677326301083, 7.71786567822794458414042360446, 8.539438678049514695525480060779, 9.216277435771475520346104219801, 10.46407485890895257777463635268, 11.08494937355173958041112882120

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