Properties

Label 2-245-7.2-c3-0-3
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  + (−0.322 + 0.558i)2-s + (−2.09 − 3.62i)3-s + (3.79 + 6.56i)4-s + (−2.5 + 4.33i)5-s + 2.69·6-s − 10.0·8-s + (4.73 − 8.20i)9-s + (−1.61 − 2.79i)10-s + (23.8 + 41.3i)11-s + (15.8 − 27.5i)12-s − 57.2·13-s + 20.9·15-s + (−27.0 + 46.9i)16-s + (−18.4 − 32.0i)17-s + (3.05 + 5.28i)18-s + (15.3 − 26.6i)19-s + ⋯
L(s)  = 1  + (−0.113 + 0.197i)2-s + (−0.402 − 0.697i)3-s + (0.474 + 0.821i)4-s + (−0.223 + 0.387i)5-s + 0.183·6-s − 0.443·8-s + (0.175 − 0.303i)9-s + (−0.0509 − 0.0882i)10-s + (0.653 + 1.13i)11-s + (0.381 − 0.661i)12-s − 1.22·13-s + 0.360·15-s + (−0.423 + 0.733i)16-s + (−0.263 − 0.456i)17-s + (0.0399 + 0.0692i)18-s + (0.185 − 0.321i)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: $\chi_{245} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 245,\ (\ :3/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.213251 + 0.694020i\)
\(L(\frac12)\) \(\approx\) \(0.213251 + 0.694020i\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.322 - 0.558i)T + (-4 - 6.92i)T^{2} \)
3 \( 1 + (2.09 + 3.62i)T + (-13.5 + 23.3i)T^{2} \)
11 \( 1 + (-23.8 - 41.3i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 57.2T + 2.19e3T^{2} \)
17 \( 1 + (18.4 + 32.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-15.3 + 26.6i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (26.5 - 46.0i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 195.T + 2.43e4T^{2} \)
31 \( 1 + (-128. - 223. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (173. - 300. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 267.T + 6.89e4T^{2} \)
43 \( 1 + 176.T + 7.95e4T^{2} \)
47 \( 1 + (155. - 269. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-246. - 426. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-49.3 - 85.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-41.0 + 71.1i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (327. + 566. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 779.T + 3.57e5T^{2} \)
73 \( 1 + (414. + 718. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-384. + 666. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 613.T + 5.71e5T^{2} \)
89 \( 1 + (228. - 396. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.41e3T + 9.12e5T^{2} \)
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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

−12.03201037634684588046637225949, −11.52834889779307487690125939934, −10.08338504873613960508889413163, −9.089481904993911123256681694245, −7.66239310611095099742576996014, −7.08357122660437241459319843105, −6.46501974625411428958775400099, −4.74399667176929956123612458880, −3.30036803660627837012351189600, −1.85794373429656740065576037771, 0.29653635153948716689887677092, 2.02046311765659396504088012331, 3.83484434926641071336630820642, 5.09680381506419919301926450177, 5.89946986034776665745862708722, 7.16361730301230471341094746393, 8.509975807529397289023081166168, 9.652969160538437223077981728839, 10.26332451571144452078695894139, 11.28493399250347407893742938297

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