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Properties

Label 2-980-140.47-c0-0-3
Degree $2$
Conductor $980$
Sign $0.343 + 0.939i$
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  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.608 − 0.793i)5-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.793 − 0.608i)10-s + (−0.541 − 0.541i)13-s + (0.500 − 0.866i)16-s + (1.78 + 0.478i)17-s + (−0.965 − 0.258i)18-s + (−0.923 − 0.382i)20-s + (−0.258 + 0.965i)25-s + (−0.662 − 0.382i)26-s + 1.41i·29-s + (0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (0.965 − 0.258i)2-s + (0.866 − 0.499i)4-s + (−0.608 − 0.793i)5-s + (0.707 − 0.707i)8-s + (−0.866 − 0.5i)9-s + (−0.793 − 0.608i)10-s + (−0.541 − 0.541i)13-s + (0.500 − 0.866i)16-s + (1.78 + 0.478i)17-s + (−0.965 − 0.258i)18-s + (−0.923 − 0.382i)20-s + (−0.258 + 0.965i)25-s + (−0.662 − 0.382i)26-s + 1.41i·29-s + (0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.343 + 0.939i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (607, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.343 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.583853583\)
\(L(\frac12)\) \(\approx\) \(1.583853583\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.608 + 0.793i)T \)
7 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
17 \( 1 + (-1.78 - 0.478i)T + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 - 1.41iT - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.866 - 0.5i)T^{2} \)
41 \( 1 - 1.84iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.662 - 0.382i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (-0.478 + 1.78i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.382 - 0.662i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
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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

−10.15382030094722538800779980884, −9.322560593405125326896884976313, −8.193648297971926298129883254836, −7.57919981732377648073048131455, −6.38336891663706686820250221749, −5.46709046308061079730720660431, −4.88225579920533243490948016353, −3.63761042511524958542888036053, −3.00806593942417746644569129180, −1.26534666742680647653959682610, 2.32359592202046439146272450793, 3.17579875706201083312047073997, 4.10891353348607409378749799953, 5.21917996317546242269317612240, 5.96396865683799723205719766856, 6.99351473109416707750755437235, 7.67796263927387445920140996231, 8.302198426001099998127953324450, 9.699820649890286765067301300057, 10.59979030326285674295559987697

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