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Properties

Label 2-980-20.3-c1-0-80
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: $\chi_{980} (883, \cdot )$
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

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

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