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Properties

Label 2-490-35.27-c1-0-6
Degree $2$
Conductor $490$
Sign $-0.542 - 0.839i$
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  + (0.707 + 0.707i)2-s + (1.42 + 1.42i)3-s + 1.00i·4-s + (0.204 + 2.22i)5-s + 2.02i·6-s + (−0.707 + 0.707i)8-s + 1.08i·9-s + (−1.42 + 1.71i)10-s − 4.03·11-s + (−1.42 + 1.42i)12-s + (0.204 + 0.204i)13-s + (−2.89 + 3.47i)15-s − 1.00·16-s + (1.44 − 1.44i)17-s + (−0.769 + 0.769i)18-s + 6.20·19-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.825 + 0.825i)3-s + 0.500i·4-s + (0.0916 + 0.995i)5-s + 0.825i·6-s + (−0.250 + 0.250i)8-s + 0.362i·9-s + (−0.452 + 0.543i)10-s − 1.21·11-s + (−0.412 + 0.412i)12-s + (0.0568 + 0.0568i)13-s + (−0.746 + 0.897i)15-s − 0.250·16-s + (0.349 − 0.349i)17-s + (−0.181 + 0.181i)18-s + 1.42·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.542 - 0.839i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(490\)    =    \(2 \cdot 5 \cdot 7^{2}\)
Sign: $-0.542 - 0.839i$
Analytic conductor: \(3.91266\)
Root analytic conductor: \(1.97804\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{490} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 490,\ (\ :1/2),\ -0.542 - 0.839i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09374 + 2.00889i\)
\(L(\frac12)\) \(\approx\) \(1.09374 + 2.00889i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.204 - 2.22i)T \)
7 \( 1 \)
good3 \( 1 + (-1.42 - 1.42i)T + 3iT^{2} \)
11 \( 1 + 4.03T + 11T^{2} \)
13 \( 1 + (-0.204 - 0.204i)T + 13iT^{2} \)
17 \( 1 + (-1.44 + 1.44i)T - 17iT^{2} \)
19 \( 1 - 6.20T + 19T^{2} \)
23 \( 1 + (-3.20 + 3.20i)T - 23iT^{2} \)
29 \( 1 + 7.15iT - 29T^{2} \)
31 \( 1 - 7.31iT - 31T^{2} \)
37 \( 1 + (-3.27 - 3.27i)T + 37iT^{2} \)
41 \( 1 - 2.58iT - 41T^{2} \)
43 \( 1 + (4.97 - 4.97i)T - 43iT^{2} \)
47 \( 1 + (0.222 - 0.222i)T - 47iT^{2} \)
53 \( 1 + (-5.85 + 5.85i)T - 53iT^{2} \)
59 \( 1 + 0.855T + 59T^{2} \)
61 \( 1 - 6.92iT - 61T^{2} \)
67 \( 1 + (2.23 + 2.23i)T + 67iT^{2} \)
71 \( 1 - 7.12T + 71T^{2} \)
73 \( 1 + (8.15 + 8.15i)T + 73iT^{2} \)
79 \( 1 - 5.07iT - 79T^{2} \)
83 \( 1 + (3.85 + 3.85i)T + 83iT^{2} \)
89 \( 1 - 3.07T + 89T^{2} \)
97 \( 1 + (6.63 - 6.63i)T - 97iT^{2} \)
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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

−11.17832380031520018781297370492, −10.16854819423309212234967147876, −9.632048901853979394809016663570, −8.433239406596272078829423207949, −7.65034206424659176125337277536, −6.72128607918530071957573171037, −5.54459176729655763351466473271, −4.53481062947817456681044568427, −3.24391210412703905596008485411, −2.76432225100279497959633066689, 1.20603263692282532759071027584, 2.41935403177018642446999542105, 3.53059743841958877120603826015, 5.01949899403594998225702658153, 5.64148136587918857402891622833, 7.25844358564364787798009516291, 7.915907207882145924844606424933, 8.847700872898961051533853511664, 9.697561716042693909641085313449, 10.73287177496011890494250351374

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