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Properties

Label 2-980-980.319-c0-0-1
Degree $2$
Conductor $980$
Sign $-0.918 + 0.394i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.955 − 0.294i)2-s + (−0.722 − 1.84i)3-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−1.23 − 1.54i)6-s + (0.826 − 0.563i)7-s + (0.623 − 0.781i)8-s + (−2.13 + 1.98i)9-s + (−0.988 + 0.149i)10-s + (−1.63 − 1.11i)12-s + (0.623 − 0.781i)14-s + (0.440 + 1.92i)15-s + (0.365 − 0.930i)16-s + (−1.45 + 2.52i)18-s + (−0.900 + 0.433i)20-s + (−1.63 − 1.11i)21-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (−0.722 − 1.84i)3-s + (0.826 − 0.563i)4-s + (−0.988 − 0.149i)5-s + (−1.23 − 1.54i)6-s + (0.826 − 0.563i)7-s + (0.623 − 0.781i)8-s + (−2.13 + 1.98i)9-s + (−0.988 + 0.149i)10-s + (−1.63 − 1.11i)12-s + (0.623 − 0.781i)14-s + (0.440 + 1.92i)15-s + (0.365 − 0.930i)16-s + (−1.45 + 2.52i)18-s + (−0.900 + 0.433i)20-s + (−1.63 − 1.11i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.918 + 0.394i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ -0.918 + 0.394i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.249764435\)
\(L(\frac12)\) \(\approx\) \(1.249764435\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.955 + 0.294i)T \)
5 \( 1 + (0.988 + 0.149i)T \)
7 \( 1 + (-0.826 + 0.563i)T \)
good3 \( 1 + (0.722 + 1.84i)T + (-0.733 + 0.680i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.0546 - 0.728i)T + (-0.988 + 0.149i)T^{2} \)
29 \( 1 + (0.134 - 0.0648i)T + (0.623 - 0.781i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.365 - 0.930i)T^{2} \)
41 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (0.914 + 1.14i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-1.19 + 0.367i)T + (0.826 - 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.826 - 0.563i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \)
89 \( 1 + (0.109 - 0.101i)T + (0.0747 - 0.997i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.43071615673202754532148423868, −8.574983847324653016088783745481, −7.72103193340108346249479932561, −7.24913875304882174491071761815, −6.51292289969412086512203631369, −5.44440505171647730946890844315, −4.76387151345814954211108831636, −3.45883999007226552796610097320, −2.07411836086762888200520563839, −1.03165305805936878369777934880, 2.80739786851157869906779037086, 3.79386229924272826275677446921, 4.48263557991472466367729611400, 5.12304470499979688468404338825, 5.89515585546760770781378637334, 6.93623835828513267878349046811, 8.228642111609071848274997737785, 8.763220285447480446073732791306, 9.997684377987310513323991221074, 10.92968838256198911991182437684

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