Properties

Label 2-980-5.4-c1-0-3
Degree $2$
Conductor $980$
Sign $-0.732 - 0.680i$
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  + 1.73i·3-s + (1.63 + 1.52i)5-s − 5.27·11-s + 2.62i·13-s + (−2.63 + 2.83i)15-s − 0.418i·17-s + 3.27·19-s + 7.82i·23-s + (0.362 + 4.98i)25-s + 5.19i·27-s − 4.27·29-s − 3.27·31-s − 9.13i·33-s − 9.97i·37-s − 4.54·39-s + ⋯
L(s)  = 1  + 0.999i·3-s + (0.732 + 0.680i)5-s − 1.59·11-s + 0.728i·13-s + (−0.680 + 0.732i)15-s − 0.101i·17-s + 0.751·19-s + 1.63i·23-s + (0.0725 + 0.997i)25-s + 1.00i·27-s − 0.793·29-s − 0.588·31-s − 1.59i·33-s − 1.63i·37-s − 0.728·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.732 - 0.680i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (589, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.732 - 0.680i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.546264 + 1.38958i\)
\(L(\frac12)\) \(\approx\) \(0.546264 + 1.38958i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.63 - 1.52i)T \)
7 \( 1 \)
good3 \( 1 - 1.73iT - 3T^{2} \)
11 \( 1 + 5.27T + 11T^{2} \)
13 \( 1 - 2.62iT - 13T^{2} \)
17 \( 1 + 0.418iT - 17T^{2} \)
19 \( 1 - 3.27T + 19T^{2} \)
23 \( 1 - 7.82iT - 23T^{2} \)
29 \( 1 + 4.27T + 29T^{2} \)
31 \( 1 + 3.27T + 31T^{2} \)
37 \( 1 + 9.97iT - 37T^{2} \)
41 \( 1 - 3.72T + 41T^{2} \)
43 \( 1 - 2.15iT - 43T^{2} \)
47 \( 1 + 6.50iT - 47T^{2} \)
53 \( 1 - 5.67iT - 53T^{2} \)
59 \( 1 + 3.27T + 59T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 3.52iT - 67T^{2} \)
71 \( 1 + 4.54T + 71T^{2} \)
73 \( 1 - 6.50iT - 73T^{2} \)
79 \( 1 + 7.27T + 79T^{2} \)
83 \( 1 + 7.40iT - 83T^{2} \)
89 \( 1 + 7T + 89T^{2} \)
97 \( 1 - 6.92iT - 97T^{2} \)
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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

−10.23526780595824844845959393662, −9.606647125399777283402477293828, −9.039413915953197922333230846196, −7.60352596594479196918412155374, −7.12266627363024209208745554155, −5.64157732044416580378530754101, −5.31589916978674769057325598316, −4.03944148425413741050640058134, −3.09914533081311057246613997630, −1.94149152314632860883411906680, 0.67009093197832453107159192149, 1.97147768849012565373514973959, 2.91229851661542080492586322657, 4.61451230414475539292835242962, 5.43388114629236229923074894758, 6.21704850716647199715687633832, 7.25394571343315023491584726957, 8.001113630154357454631438207766, 8.619801764797419695832760086512, 9.844755192134472921899048714148

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