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Properties

Label 2-980-140.139-c1-0-32
Degree $2$
Conductor $980$
Sign $-0.0335 - 0.999i$
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  + (−1.36 + 0.385i)2-s + 0.423i·3-s + (1.70 − 1.04i)4-s + (1.74 + 1.39i)5-s + (−0.163 − 0.576i)6-s + (−1.91 + 2.08i)8-s + 2.82·9-s + (−2.91 − 1.23i)10-s + 4.89i·11-s + (0.443 + 0.721i)12-s + 2.54·13-s + (−0.591 + 0.738i)15-s + (1.80 − 3.57i)16-s − 5.11·17-s + (−3.83 + 1.08i)18-s + 6.26·19-s + ⋯
L(s)  = 1  + (−0.962 + 0.272i)2-s + 0.244i·3-s + (0.851 − 0.524i)4-s + (0.780 + 0.625i)5-s + (−0.0665 − 0.235i)6-s + (−0.676 + 0.736i)8-s + 0.940·9-s + (−0.921 − 0.388i)10-s + 1.47i·11-s + (0.128 + 0.208i)12-s + 0.706·13-s + (−0.152 + 0.190i)15-s + (0.450 − 0.892i)16-s − 1.24·17-s + (−0.904 + 0.256i)18-s + 1.43·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0335 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0335 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0335 - 0.999i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.0335 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.873939 + 0.903739i\)
\(L(\frac12)\) \(\approx\) \(0.873939 + 0.903739i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 - 0.385i)T \)
5 \( 1 + (-1.74 - 1.39i)T \)
7 \( 1 \)
good3 \( 1 - 0.423iT - 3T^{2} \)
11 \( 1 - 4.89iT - 11T^{2} \)
13 \( 1 - 2.54T + 13T^{2} \)
17 \( 1 + 5.11T + 17T^{2} \)
19 \( 1 - 6.26T + 19T^{2} \)
23 \( 1 + 4.63T + 23T^{2} \)
29 \( 1 + 1.88T + 29T^{2} \)
31 \( 1 + 1.47T + 31T^{2} \)
37 \( 1 - 2.35iT - 37T^{2} \)
41 \( 1 + 7.05iT - 41T^{2} \)
43 \( 1 - 10.7T + 43T^{2} \)
47 \( 1 + 12.2iT - 47T^{2} \)
53 \( 1 - 2.23iT - 53T^{2} \)
59 \( 1 + 5.68T + 59T^{2} \)
61 \( 1 - 3.87iT - 61T^{2} \)
67 \( 1 + 0.889T + 67T^{2} \)
71 \( 1 - 14.3iT - 71T^{2} \)
73 \( 1 + 7.87T + 73T^{2} \)
79 \( 1 - 4.63iT - 79T^{2} \)
83 \( 1 - 4.32iT - 83T^{2} \)
89 \( 1 + 2.18iT - 89T^{2} \)
97 \( 1 - 3.42T + 97T^{2} \)
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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

−10.05184807462169006539803573940, −9.525720080845937124524649598245, −8.788963126707178725299088630041, −7.40208583894422020001389340684, −7.12984292618136733836354877023, −6.16734502263099378833724428411, −5.20785998500267038651613858071, −3.94653126336625394791384742158, −2.40694537702422390976874698185, −1.55115276311505078982875549075, 0.853044336688105131540031543319, 1.82564796081963552995737426588, 3.12892807189809310164980063130, 4.34978878799297992224456288488, 5.84859617521673763669525145826, 6.33678985323859092000106994561, 7.50715412446562702709687554763, 8.241057642095467474362910856091, 9.151473384117175381132148060550, 9.528805821842074310998471900524

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