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Properties

Label 2-980-140.87-c0-0-1
Degree $2$
Conductor $980$
Sign $0.703 + 0.710i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.130 + 0.991i)5-s + (−0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (0.991 + 0.130i)10-s + (1.30 + 1.30i)13-s + (0.500 + 0.866i)16-s + (−0.198 − 0.739i)17-s + (−0.258 − 0.965i)18-s + (0.382 − 0.923i)20-s + (−0.965 + 0.258i)25-s + (1.60 − 0.923i)26-s − 1.41i·29-s + (0.965 − 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.130 + 0.991i)5-s + (−0.707 + 0.707i)8-s + (0.866 − 0.5i)9-s + (0.991 + 0.130i)10-s + (1.30 + 1.30i)13-s + (0.500 + 0.866i)16-s + (−0.198 − 0.739i)17-s + (−0.258 − 0.965i)18-s + (0.382 − 0.923i)20-s + (−0.965 + 0.258i)25-s + (1.60 − 0.923i)26-s − 1.41i·29-s + (0.965 − 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.703 + 0.710i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.703 + 0.710i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.157307528\)
\(L(\frac12)\) \(\approx\) \(1.157307528\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (-0.130 - 0.991i)T \)
7 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.30 - 1.30i)T + iT^{2} \)
17 \( 1 + (0.198 + 0.739i)T + (-0.866 + 0.5i)T^{2} \)
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 \( 1 + 0.765iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (1.60 - 0.923i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.739 - 0.198i)T + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + (0.923 + 1.60i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (1.30 - 1.30i)T - iT^{2} \)
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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

−10.21352353197693310844521312656, −9.449845463860393548175377977374, −8.819102179855448933235297364697, −7.50092223819992485760194973549, −6.55366923951881129783280963285, −5.86300723368896166762589312285, −4.38216827304801086991878245050, −3.82222009040272854744947966976, −2.67855138154227589989887985242, −1.52419860039426188853104628960, 1.35823294481175125934541260987, 3.38784059187083240418936999218, 4.33917652896575263793884873100, 5.20736999260837875625028046869, 5.91899642653050994539631344398, 6.88707422233749148537472870336, 8.000041644299861961281266752493, 8.351690510411509020964212046110, 9.240941547246737477184807371208, 10.15892248415177991666600029348

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