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Properties

Label 2-490-245.108-c1-0-13
Degree $2$
Conductor $490$
Sign $0.843 + 0.537i$
Analytic cond. $3.91266$
Root an. cond. $1.97804$
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  + (−0.804 + 0.593i)2-s + (−0.619 + 3.27i)3-s + (0.294 − 0.955i)4-s + (−1.61 − 1.54i)5-s + (−1.44 − 3.00i)6-s + (2.64 − 0.124i)7-s + (0.330 + 0.943i)8-s + (−7.53 − 2.95i)9-s + (2.21 + 0.287i)10-s + (−0.958 − 2.44i)11-s + (2.94 + 1.55i)12-s + (−0.567 − 5.04i)13-s + (−2.05 + 1.66i)14-s + (6.06 − 4.32i)15-s + (−0.826 − 0.563i)16-s + (−0.149 − 3.99i)17-s + ⋯
L(s)  = 1  + (−0.568 + 0.419i)2-s + (−0.357 + 1.88i)3-s + (0.147 − 0.477i)4-s + (−0.721 − 0.692i)5-s + (−0.589 − 1.22i)6-s + (0.998 − 0.0469i)7-s + (0.116 + 0.333i)8-s + (−2.51 − 0.985i)9-s + (0.701 + 0.0908i)10-s + (−0.288 − 0.735i)11-s + (0.850 + 0.449i)12-s + (−0.157 − 1.39i)13-s + (−0.548 + 0.446i)14-s + (1.56 − 1.11i)15-s + (−0.206 − 0.140i)16-s + (−0.0362 − 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.843 + 0.537i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ 0.843 + 0.537i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453116 - 0.132032i\)
\(L(\frac12)\) \(\approx\) \(0.453116 - 0.132032i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.804 - 0.593i)T \)
5 \( 1 + (1.61 + 1.54i)T \)
7 \( 1 + (-2.64 + 0.124i)T \)
good3 \( 1 + (0.619 - 3.27i)T + (-2.79 - 1.09i)T^{2} \)
11 \( 1 + (0.958 + 2.44i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (0.567 + 5.04i)T + (-12.6 + 2.89i)T^{2} \)
17 \( 1 + (0.149 + 3.99i)T + (-16.9 + 1.27i)T^{2} \)
19 \( 1 + (0.388 + 0.673i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.02 + 0.0758i)T + (22.9 + 1.71i)T^{2} \)
29 \( 1 + (7.84 + 1.79i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (0.350 + 0.202i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.68 - 5.07i)T + (-20.8 - 30.5i)T^{2} \)
41 \( 1 + (0.312 - 0.648i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (-6.78 - 2.37i)T + (33.6 + 26.8i)T^{2} \)
47 \( 1 + (1.03 + 1.40i)T + (-13.8 + 44.9i)T^{2} \)
53 \( 1 + (-2.21 - 4.18i)T + (-29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.244 - 3.26i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (4.45 + 14.4i)T + (-50.4 + 34.3i)T^{2} \)
67 \( 1 + (2.47 + 9.24i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.244 + 1.07i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.16 + 2.93i)T + (-21.5 - 69.7i)T^{2} \)
79 \( 1 + (10.1 - 5.88i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.31 + 0.711i)T + (80.9 + 18.4i)T^{2} \)
89 \( 1 + (-2.57 + 6.56i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (-9.33 - 9.33i)T + 97iT^{2} \)
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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

−10.94452839819065087387854690591, −9.928909346995847475137290696112, −9.108236219794911598580343218570, −8.343114374891184931201671164377, −7.67045525648009814765513518698, −5.74896769611334139614271823099, −5.18849955376287881695872370826, −4.37892524343925046978608854015, −3.17047850117770146183189479697, −0.35680662015563805610690938868, 1.63395275785729967993995787762, 2.33659133476380863240858715810, 4.11203113895699004381329687338, 5.73340935034642251693674344922, 6.92901546763162283444331839926, 7.37847045423344990086357610442, 8.097175805249357288186957565725, 8.945008951343313981792922937358, 10.57077928536409487817479377445, 11.24397859413288724567644874086

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