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Properties

Label 2-980-20.7-c1-0-37
Degree $2$
Conductor $980$
Sign $-0.357 - 0.934i$
Analytic cond. $7.82533$
Root an. cond. $2.79738$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.121 + 1.40i)2-s + (0.404 + 0.404i)3-s + (−1.97 − 0.341i)4-s + (2.23 + 0.0378i)5-s + (−0.618 + 0.520i)6-s + (0.719 − 2.73i)8-s − 2.67i·9-s + (−0.323 + 3.14i)10-s + 3.96i·11-s + (−0.658 − 0.934i)12-s + (−2.04 + 2.04i)13-s + (0.888 + 0.918i)15-s + (3.76 + 1.34i)16-s + (0.424 + 0.424i)17-s + (3.76 + 0.323i)18-s + 2.00·19-s + ⋯
L(s)  = 1  + (−0.0855 + 0.996i)2-s + (0.233 + 0.233i)3-s + (−0.985 − 0.170i)4-s + (0.999 + 0.0169i)5-s + (−0.252 + 0.212i)6-s + (0.254 − 0.967i)8-s − 0.891i·9-s + (−0.102 + 0.994i)10-s + 1.19i·11-s + (−0.190 − 0.269i)12-s + (−0.568 + 0.568i)13-s + (0.229 + 0.237i)15-s + (0.941 + 0.336i)16-s + (0.103 + 0.103i)17-s + (0.887 + 0.0762i)18-s + 0.460·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.357 - 0.934i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.978420 + 1.42174i\)
\(L(\frac12)\) \(\approx\) \(0.978420 + 1.42174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.121 - 1.40i)T \)
5 \( 1 + (-2.23 - 0.0378i)T \)
7 \( 1 \)
good3 \( 1 + (-0.404 - 0.404i)T + 3iT^{2} \)
11 \( 1 - 3.96iT - 11T^{2} \)
13 \( 1 + (2.04 - 2.04i)T - 13iT^{2} \)
17 \( 1 + (-0.424 - 0.424i)T + 17iT^{2} \)
19 \( 1 - 2.00T + 19T^{2} \)
23 \( 1 + (-1.75 - 1.75i)T + 23iT^{2} \)
29 \( 1 - 7.25iT - 29T^{2} \)
31 \( 1 - 3.26iT - 31T^{2} \)
37 \( 1 + (-6.80 - 6.80i)T + 37iT^{2} \)
41 \( 1 - 3.35T + 41T^{2} \)
43 \( 1 + (-3.31 - 3.31i)T + 43iT^{2} \)
47 \( 1 + (-8.86 + 8.86i)T - 47iT^{2} \)
53 \( 1 + (2.12 - 2.12i)T - 53iT^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 + 4.23T + 61T^{2} \)
67 \( 1 + (2.68 - 2.68i)T - 67iT^{2} \)
71 \( 1 + 4.86iT - 71T^{2} \)
73 \( 1 + (-6.91 + 6.91i)T - 73iT^{2} \)
79 \( 1 - 6.91T + 79T^{2} \)
83 \( 1 + (5.46 + 5.46i)T + 83iT^{2} \)
89 \( 1 + 3.87iT - 89T^{2} \)
97 \( 1 + (1.11 + 1.11i)T + 97iT^{2} \)
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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

−9.801476631417685299258892315605, −9.403965000194317568114934045994, −8.802454134473266135215421437807, −7.52939910908588438254849577882, −6.86661291899947854219791584025, −6.11551915294422656938463509695, −5.09593865106545928255610225948, −4.37011100787168471523106196981, −3.03582029637693706604492141609, −1.43129120536541398692883765858, 0.906087886446534751278127850033, 2.33452959330129582034022622439, 2.88691688700947802100905880844, 4.31265421969486420034013013092, 5.38605206518459408366700457188, 5.99380709926250910020888558389, 7.56280679286536878232615099886, 8.180125412511852686617640455866, 9.204008047627337615976129882244, 9.700365078450885074973835478335

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