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Properties

Label 2-490-5.4-c1-0-17
Degree $2$
Conductor $490$
Sign $-0.894 + 0.447i$
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  + i·2-s − 3i·3-s − 4-s + (−1 − 2i)5-s + 3·6-s − i·8-s − 6·9-s + (2 − i)10-s + 3i·12-s − 2i·13-s + (−6 + 3i)15-s + 16-s + 2i·17-s − 6i·18-s − 2·19-s + (1 + 2i)20-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.73i·3-s − 0.5·4-s + (−0.447 − 0.894i)5-s + 1.22·6-s − 0.353i·8-s − 2·9-s + (0.632 − 0.316i)10-s + 0.866i·12-s − 0.554i·13-s + (−1.54 + 0.774i)15-s + 0.250·16-s + 0.485i·17-s − 1.41i·18-s − 0.458·19-s + (0.223 + 0.447i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.183384 - 0.776830i\)
\(L(\frac12)\) \(\approx\) \(0.183384 - 0.776830i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
5 \( 1 + (1 + 2i)T \)
7 \( 1 \)
good3 \( 1 + 3iT - 3T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - iT - 23T^{2} \)
29 \( 1 - T + 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 + 5iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 2T + 59T^{2} \)
61 \( 1 - 9T + 61T^{2} \)
67 \( 1 + 7iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 10iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 9iT - 83T^{2} \)
89 \( 1 + 7T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.74763762730017023390043941011, −9.179472810046171780248909079025, −8.476055642707823217919944216769, −7.71749910702203284772542943788, −7.12006120424806721003646622153, −6.01428805669115130069272882460, −5.27117429103822672790357983891, −3.74464901646632227273354820514, −1.92285065984731881074930943091, −0.47493605841542549925847141317, 2.59266321899983891570887367014, 3.60656505690847192980830180824, 4.32131888863180932911347754335, 5.33256647568010760834851226251, 6.64714120899035296149152315967, 8.039708844602938549015913210554, 9.065063327136548304541031009421, 9.767877890927488897866430958994, 10.49506508414703429801356897994, 11.20625310417646513588492080102

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