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Properties

Label 2-490-245.132-c1-0-1
Degree $2$
Conductor $490$
Sign $-0.366 - 0.930i$
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  + (0.846 − 0.532i)2-s + (−2.67 + 0.301i)3-s + (0.433 − 0.900i)4-s + (−2.10 − 0.754i)5-s + (−2.10 + 1.67i)6-s + (1.62 − 2.08i)7-s + (−0.111 − 0.993i)8-s + (4.15 − 0.948i)9-s + (−2.18 + 0.480i)10-s + (−0.835 + 3.65i)11-s + (−0.889 + 2.54i)12-s + (−2.23 + 1.40i)13-s + (0.267 − 2.63i)14-s + (5.86 + 1.38i)15-s + (−0.623 − 0.781i)16-s + (−1.33 + 3.81i)17-s + ⋯
L(s)  = 1  + (0.598 − 0.376i)2-s + (−1.54 + 0.174i)3-s + (0.216 − 0.450i)4-s + (−0.941 − 0.337i)5-s + (−0.860 + 0.685i)6-s + (0.614 − 0.788i)7-s + (−0.0395 − 0.351i)8-s + (1.38 − 0.316i)9-s + (−0.690 + 0.152i)10-s + (−0.251 + 1.10i)11-s + (−0.256 + 0.734i)12-s + (−0.619 + 0.389i)13-s + (0.0714 − 0.703i)14-s + (1.51 + 0.357i)15-s + (−0.155 − 0.195i)16-s + (−0.323 + 0.925i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.366 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.366 - 0.930i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.366 - 0.930i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.148030 + 0.217442i\)
\(L(\frac12)\) \(\approx\) \(0.148030 + 0.217442i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.846 + 0.532i)T \)
5 \( 1 + (2.10 + 0.754i)T \)
7 \( 1 + (-1.62 + 2.08i)T \)
good3 \( 1 + (2.67 - 0.301i)T + (2.92 - 0.667i)T^{2} \)
11 \( 1 + (0.835 - 3.65i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (2.23 - 1.40i)T + (5.64 - 11.7i)T^{2} \)
17 \( 1 + (1.33 - 3.81i)T + (-13.2 - 10.5i)T^{2} \)
19 \( 1 + 4.32T + 19T^{2} \)
23 \( 1 + (5.05 - 1.76i)T + (17.9 - 14.3i)T^{2} \)
29 \( 1 + (-3.31 - 6.89i)T + (-18.0 + 22.6i)T^{2} \)
31 \( 1 - 2.35iT - 31T^{2} \)
37 \( 1 + (4.20 + 1.47i)T + (28.9 + 23.0i)T^{2} \)
41 \( 1 + (-2.99 - 2.39i)T + (9.12 + 39.9i)T^{2} \)
43 \( 1 + (6.43 + 0.724i)T + (41.9 + 9.56i)T^{2} \)
47 \( 1 + (4.83 + 7.69i)T + (-20.3 + 42.3i)T^{2} \)
53 \( 1 + (-2.10 + 0.736i)T + (41.4 - 33.0i)T^{2} \)
59 \( 1 + (-5.53 - 6.93i)T + (-13.1 + 57.5i)T^{2} \)
61 \( 1 + (-2.11 - 4.39i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (4.26 + 4.26i)T + 67iT^{2} \)
71 \( 1 + (10.2 + 4.93i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-8.12 + 12.9i)T + (-31.6 - 65.7i)T^{2} \)
79 \( 1 + 0.691iT - 79T^{2} \)
83 \( 1 + (7.23 - 11.5i)T + (-36.0 - 74.7i)T^{2} \)
89 \( 1 + (-2.57 - 11.2i)T + (-80.1 + 38.6i)T^{2} \)
97 \( 1 + (-1.95 + 1.95i)T - 97iT^{2} \)
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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

−11.28837104244552094132061193491, −10.60675360219785854908532628094, −10.05107354728169954291195640846, −8.454127547871175524043926399066, −7.26060969335788772991325794991, −6.59318082175203919161071582520, −5.22747580701635828762422704237, −4.59488827581110387807160190840, −3.93781647443439673846093240523, −1.65458583549934365454718145742, 0.15599171587765867162665321445, 2.65594982180106610380472383822, 4.26467581343651269785601556232, 5.09818444958067084745270886366, 5.96446258045163683869271217833, 6.69377541898525672845473559548, 7.82010166400934118087653740648, 8.513086218033490161252501049331, 10.19034273528579332313266896455, 11.17668066343998610940884654837

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