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Properties

Label 2-490-245.108-c1-0-10
Degree $2$
Conductor $490$
Sign $0.690 - 0.723i$
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 + (−0.312 + 1.65i)3-s + (0.294 − 0.955i)4-s + (2.21 + 0.320i)5-s + (0.729 + 1.51i)6-s + (−2.41 + 1.08i)7-s + (−0.330 − 0.943i)8-s + (0.159 + 0.0625i)9-s + (1.97 − 1.05i)10-s + (1.58 + 4.03i)11-s + (1.48 + 0.785i)12-s + (0.262 + 2.32i)13-s + (−1.29 + 2.30i)14-s + (−1.22 + 3.55i)15-s + (−0.826 − 0.563i)16-s + (−0.0291 − 0.779i)17-s + ⋯
L(s)  = 1  + (0.568 − 0.419i)2-s + (−0.180 + 0.954i)3-s + (0.147 − 0.477i)4-s + (0.989 + 0.143i)5-s + (0.297 + 0.618i)6-s + (−0.911 + 0.411i)7-s + (−0.116 − 0.333i)8-s + (0.0531 + 0.0208i)9-s + (0.623 − 0.333i)10-s + (0.477 + 1.21i)11-s + (0.429 + 0.226i)12-s + (0.0727 + 0.645i)13-s + (−0.345 + 0.616i)14-s + (−0.315 + 0.918i)15-s + (−0.206 − 0.140i)16-s + (−0.00707 − 0.189i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $0.690 - 0.723i$
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: \((2,\ 490,\ (\ :1/2),\ 0.690 - 0.723i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84840 + 0.790529i\)
\(L(\frac12)\) \(\approx\) \(1.84840 + 0.790529i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.21 - 0.320i)T \)
7 \( 1 + (2.41 - 1.08i)T \)
good3 \( 1 + (0.312 - 1.65i)T + (-2.79 - 1.09i)T^{2} \)
11 \( 1 + (-1.58 - 4.03i)T + (-8.06 + 7.48i)T^{2} \)
13 \( 1 + (-0.262 - 2.32i)T + (-12.6 + 2.89i)T^{2} \)
17 \( 1 + (0.0291 + 0.779i)T + (-16.9 + 1.27i)T^{2} \)
19 \( 1 + (3.47 + 6.01i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.30 - 0.123i)T + (22.9 + 1.71i)T^{2} \)
29 \( 1 + (-8.83 - 2.01i)T + (26.1 + 12.5i)T^{2} \)
31 \( 1 + (2.15 + 1.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.17 - 4.10i)T + (-20.8 - 30.5i)T^{2} \)
41 \( 1 + (-0.857 + 1.78i)T + (-25.5 - 32.0i)T^{2} \)
43 \( 1 + (1.34 + 0.471i)T + (33.6 + 26.8i)T^{2} \)
47 \( 1 + (-0.539 - 0.730i)T + (-13.8 + 44.9i)T^{2} \)
53 \( 1 + (4.74 + 8.97i)T + (-29.8 + 43.7i)T^{2} \)
59 \( 1 + (0.327 - 4.36i)T + (-58.3 - 8.79i)T^{2} \)
61 \( 1 + (2.74 + 8.88i)T + (-50.4 + 34.3i)T^{2} \)
67 \( 1 + (3.49 + 13.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (0.703 + 3.08i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-0.914 + 1.23i)T + (-21.5 - 69.7i)T^{2} \)
79 \( 1 + (-9.34 + 5.39i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.11 + 0.463i)T + (80.9 + 18.4i)T^{2} \)
89 \( 1 + (-4.82 + 12.2i)T + (-65.2 - 60.5i)T^{2} \)
97 \( 1 + (-5.07 - 5.07i)T + 97iT^{2} \)
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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

−10.87309944484687907587915220564, −10.21721104809858836550218843157, −9.434594963059406637565850404304, −9.052462944062991022954385999386, −6.89288530846570454567001731588, −6.44542782205485892656951423270, −5.05975957524144058733248615750, −4.51979972069919769118528315134, −3.16060275978258346076370767899, −1.97671030646765030252479165401, 1.15906611716601813338575457518, 2.81432170806898429125962172028, 4.01707936310981079715936354733, 5.62167452025057283918300268492, 6.25108071868664219803705389638, 6.79601259769603891171759102832, 7.978069061112443235182134967666, 8.892468469696880142961959181672, 10.04975292036761349152143780306, 10.81833599456831268174369289439

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