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Properties

Label 2-980-980.179-c0-0-0
Degree $2$
Conductor $980$
Sign $0.926 + 0.375i$
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  + (0.0747 + 0.997i)2-s + (−1.40 − 0.432i)3-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (0.326 − 1.42i)6-s + (−0.988 + 0.149i)7-s + (−0.222 − 0.974i)8-s + (0.949 + 0.647i)9-s + (−0.733 − 0.680i)10-s + (1.44 + 0.218i)12-s + (−0.222 − 0.974i)14-s + (1.32 − 0.636i)15-s + (0.955 − 0.294i)16-s + (−0.574 + 0.995i)18-s + (0.623 − 0.781i)20-s + (1.44 + 0.218i)21-s + ⋯
L(s)  = 1  + (0.0747 + 0.997i)2-s + (−1.40 − 0.432i)3-s + (−0.988 + 0.149i)4-s + (−0.733 + 0.680i)5-s + (0.326 − 1.42i)6-s + (−0.988 + 0.149i)7-s + (−0.222 − 0.974i)8-s + (0.949 + 0.647i)9-s + (−0.733 − 0.680i)10-s + (1.44 + 0.218i)12-s + (−0.222 − 0.974i)14-s + (1.32 − 0.636i)15-s + (0.955 − 0.294i)16-s + (−0.574 + 0.995i)18-s + (0.623 − 0.781i)20-s + (1.44 + 0.218i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 + 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.926 + 0.375i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.926 + 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2822460454\)
\(L(\frac12)\) \(\approx\) \(0.2822460454\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (0.733 - 0.680i)T \)
7 \( 1 + (0.988 - 0.149i)T \)
good3 \( 1 + (1.40 + 0.432i)T + (0.826 + 0.563i)T^{2} \)
11 \( 1 + (-0.365 + 0.930i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.733 - 0.680i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.698 + 1.77i)T + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (-0.455 + 0.571i)T + (-0.222 - 0.974i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.367 + 1.61i)T + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.367 - 1.61i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.0332 + 0.443i)T + (-0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (1.88 + 0.284i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.988 + 0.149i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.603 - 0.411i)T + (0.365 + 0.930i)T^{2} \)
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

−10.32566600966433427992347810546, −9.236083748263197054717716912858, −8.218522372616294898633467293454, −7.27574548511980914424108847210, −6.51051982959297572214503629437, −6.29431303694510534831611333582, −5.15603961476170398631311017871, −4.23192209226235242270059957780, −3.01893850254757872878226300448, −0.38224677670710027953203313499, 1.13289489946018210131499259401, 3.18510625457038375632263541829, 4.02984960850434376795378802722, 4.97518991908291785226321029949, 5.56969886951236791490410693732, 6.67247232636243139429814131805, 7.83292919196912417583909673061, 9.015484113181871244016368286741, 9.609888802203966425222280890599, 10.47062659900989405959857402694

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