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Properties

Label 2-980-980.739-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  + (0.826 + 0.563i)2-s + (−1.40 − 1.29i)3-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.425 − 1.86i)6-s + (0.365 + 0.930i)7-s + (−0.222 + 0.974i)8-s + (0.198 + 2.64i)9-s + (0.955 + 0.294i)10-s + (0.698 − 1.77i)12-s + (−0.222 + 0.974i)14-s + (−1.72 − 0.829i)15-s + (−0.733 + 0.680i)16-s + (−1.32 + 2.29i)18-s + (0.623 + 0.781i)20-s + (0.698 − 1.77i)21-s + ⋯
L(s)  = 1  + (0.826 + 0.563i)2-s + (−1.40 − 1.29i)3-s + (0.365 + 0.930i)4-s + (0.955 − 0.294i)5-s + (−0.425 − 1.86i)6-s + (0.365 + 0.930i)7-s + (−0.222 + 0.974i)8-s + (0.198 + 2.64i)9-s + (0.955 + 0.294i)10-s + (0.698 − 1.77i)12-s + (−0.222 + 0.974i)14-s + (−1.72 − 0.829i)15-s + (−0.733 + 0.680i)16-s + (−1.32 + 2.29i)18-s + (0.623 + 0.781i)20-s + (0.698 − 1.77i)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: $\chi_{980} (739, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.242546729\)
\(L(\frac12)\) \(\approx\) \(1.242546729\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.826 - 0.563i)T \)
5 \( 1 + (-0.955 + 0.294i)T \)
7 \( 1 + (-0.365 - 0.930i)T \)
good3 \( 1 + (1.40 + 1.29i)T + (0.0747 + 0.997i)T^{2} \)
11 \( 1 + (0.988 + 0.149i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.955 + 0.294i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-1.44 - 0.218i)T + (0.955 + 0.294i)T^{2} \)
29 \( 1 + (1.23 + 1.54i)T + (-0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.733 + 0.680i)T^{2} \)
41 \( 1 + (0.0332 - 0.145i)T + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.0332 + 0.145i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.367 + 0.250i)T + (0.365 + 0.930i)T^{2} \)
53 \( 1 + (0.733 - 0.680i)T^{2} \)
59 \( 1 + (-0.826 - 0.563i)T^{2} \)
61 \( 1 + (0.535 - 1.36i)T + (-0.733 - 0.680i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.658 + 0.317i)T + (0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.147 + 1.97i)T + (-0.988 + 0.149i)T^{2} \)
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

−10.63796553098880642628615171744, −9.258424329842795089002526145806, −8.253339492476357439523933443465, −7.40624660858818144867736546213, −6.58192866326030061269241690672, −5.85425539065709795318461966942, −5.41968920760938114852150064655, −4.65596854060343144417377938688, −2.60458097864510954174258317941, −1.67484850489566202843976501535, 1.29703237869354356287324328543, 3.16811109086690233612643194950, 4.06982302464285082845019144309, 5.06015205819469287105348343141, 5.38506159746815339783466518177, 6.47730290448584757671238249228, 7.02800535029908156528250605834, 9.113683305465989819301310113971, 9.719996175287959173538961884605, 10.48151056757867988615610766701

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