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Properties

Label 2-980-140.139-c1-0-64
Degree $2$
Conductor $980$
Sign $0.277 + 0.960i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.277 + 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.277 + 0.960i$
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: \((2,\ 980,\ (\ :1/2),\ 0.277 + 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.544129 - 0.409334i\)
\(L(\frac12)\) \(\approx\) \(0.544129 - 0.409334i\)
\(L(\frac{3}{2})\) 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

−10.04081128400024297605089400328, −8.977826709125875115199842046999, −8.157086377408563965289590255794, −7.52535347202671296701170832955, −6.70299907196099508331374890128, −5.73239694653302311297990095911, −4.08786647449685797695103002394, −3.43243088101486807484913663598, −2.20998014045916134004057994940, −0.47754250680285151547833910002, 1.22703849621438465423137230664, 2.34442292278684794566963064703, 4.09638739652273207096865611077, 4.89098561961929100151483351360, 6.16117970127656949502883840054, 7.18370266724588254408904266406, 7.66643037994739798608417070383, 8.310656328198516145698969602987, 9.522155546706164107948301508100, 9.866929588527376311037673980069

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