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Properties

Label 2-980-7.4-c1-0-12
Degree $2$
Conductor $980$
Sign $-0.701 + 0.712i$
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  + (1.5 − 2.59i)3-s + (−0.5 − 0.866i)5-s + (−3 − 5.19i)9-s + (2.5 − 4.33i)11-s + 3·13-s − 3·15-s + (−0.5 + 0.866i)17-s + (3 + 5.19i)19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s − 9·27-s − 9·29-s + (−2 + 3.46i)31-s + (−7.50 − 12.9i)33-s + (−1 − 1.73i)37-s + ⋯
L(s)  = 1  + (0.866 − 1.49i)3-s + (−0.223 − 0.387i)5-s + (−1 − 1.73i)9-s + (0.753 − 1.30i)11-s + 0.832·13-s − 0.774·15-s + (−0.121 + 0.210i)17-s + (0.688 + 1.19i)19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s − 1.73·27-s − 1.67·29-s + (−0.359 + 0.622i)31-s + (−1.30 − 2.26i)33-s + (−0.164 − 0.284i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.701 + 0.712i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.786133 - 1.87584i\)
\(L(\frac12)\) \(\approx\) \(0.786133 - 1.87584i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.5 + 4.33i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 4T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (0.5 + 0.866i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2 - 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6 - 10.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 + (-2 - 3.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 13T + 97T^{2} \)
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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

−9.170036079240335566035548870042, −8.780991723512763655898270630825, −7.984498717222100246955147740029, −7.39754411231380091219373760824, −6.22680407899718166068467139003, −5.82795065963223808062270303981, −3.95417231387729679149553435415, −3.23561328476068011249769120887, −1.86184011198517421290437146900, −0.893051422165929279927283314065, 2.06012136757807741944124969827, 3.29885871641974753156334332248, 3.99431508679137193633347955316, 4.75357227039929445424942766486, 5.85384386671571582830167338039, 7.18652279334543349240355008413, 7.84282940941575329803294748095, 9.144391342023484309781584764377, 9.314339183736443234315971302491, 10.07874805288484003330024534610

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