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Properties

Label 2-980-140.103-c0-0-0
Degree $2$
Conductor $980$
Sign $-0.869 - 0.493i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.869 - 0.493i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (803, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ -0.869 - 0.493i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9592176148\)
\(L(\frac12)\) \(\approx\) \(0.9592176148\)
\(L(1)\) 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

−10.28529435309767143971016409827, −9.703127666826961630414869059863, −8.822326162851255659638680543331, −7.62966030975571382024325477104, −7.10371619423048781112271099080, −6.65791980918712619672118067346, −5.32422079900239957500333834187, −4.56258478070144016089545331451, −3.56931751041668295086718724961, −2.26745984249529510201006063058, 0.886067886780049811381084181155, 2.25334923901345136494817088892, 3.57086415235702751703695236492, 4.45838954127245990694657927025, 5.21471178722698592948892576548, 6.14487918384568167868075484309, 7.61914742384258105904552425023, 8.280780912189857699892271367715, 9.359752575353408936399663968191, 9.889952172309343810339875382122

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