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Properties

Label 2-245-7.2-c3-0-37
Degree $2$
Conductor $245$
Sign $-0.198 - 0.980i$
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  + (2.75 − 4.77i)2-s + (−1.93 − 3.34i)3-s + (−11.2 − 19.4i)4-s + (2.5 − 4.33i)5-s − 21.3·6-s − 79.7·8-s + (6.04 − 10.4i)9-s + (−13.7 − 23.8i)10-s + (−17.2 − 29.8i)11-s + (−43.3 + 75.0i)12-s + 68.8·13-s − 19.3·15-s + (−130. + 225. i)16-s + (45.7 + 79.1i)17-s + (−33.3 − 57.8i)18-s + (−5.91 + 10.2i)19-s + ⋯
L(s)  = 1  + (0.975 − 1.68i)2-s + (−0.371 − 0.643i)3-s + (−1.40 − 2.43i)4-s + (0.223 − 0.387i)5-s − 1.44·6-s − 3.52·8-s + (0.224 − 0.388i)9-s + (−0.436 − 0.755i)10-s + (−0.473 − 0.819i)11-s + (−1.04 + 1.80i)12-s + 1.46·13-s − 0.332·15-s + (−2.03 + 3.52i)16-s + (0.652 + 1.12i)17-s + (−0.437 − 0.756i)18-s + (−0.0714 + 0.123i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(245\)    =    \(5 \cdot 7^{2}\)
Sign: $-0.198 - 0.980i$
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: \((2,\ 245,\ (\ :3/2),\ -0.198 - 0.980i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.40347 + 1.71542i\)
\(L(\frac12)\) \(\approx\) \(1.40347 + 1.71542i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.75 + 4.77i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (1.93 + 3.34i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (17.2 + 29.8i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 68.8T + 2.19e3T^{2} \)
17 \( 1 + (-45.7 - 79.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (5.91 - 10.2i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-0.0520 + 0.0902i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 190.T + 2.43e4T^{2} \)
31 \( 1 + (79.9 + 138. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-88.9 + 154. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 145.T + 6.89e4T^{2} \)
43 \( 1 - 8.25T + 7.95e4T^{2} \)
47 \( 1 + (130. - 225. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (176. + 305. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (120. + 208. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (389. - 673. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (75.9 + 131. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 311.T + 3.57e5T^{2} \)
73 \( 1 + (319. + 554. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (195. - 338. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 493.T + 5.71e5T^{2} \)
89 \( 1 + (236. - 410. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 839.T + 9.12e5T^{2} \)
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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

−11.14278763553945472528190026549, −10.47325076997247893076241793450, −9.374388320296350411756624034307, −8.312981597263185771322986058127, −6.14162294310434415898388248528, −5.77044332510232032684531154833, −4.24202430251286158887550482880, −3.25234319059377601886913148955, −1.63971458443967442282468078422, −0.72866282719264103454596716090, 3.15506161678882885871778281982, 4.43198278217124156954264416412, 5.18528474836757826789905026149, 6.16486070589528628166585859057, 7.15678798393939845402489445431, 8.028548257131460778295696322029, 9.192102324664999585120072994747, 10.33966064748472539015293784788, 11.60201141022801880962150394836, 12.71314192386163568335074497965

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