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Properties

Label 2-980-140.107-c1-0-1
Degree $2$
Conductor $980$
Sign $-0.992 - 0.123i$
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  + (0.607 + 1.27i)2-s + (0.643 − 2.40i)3-s + (−1.26 + 1.55i)4-s + (−1.36 − 1.77i)5-s + (3.45 − 0.637i)6-s + (−2.74 − 0.667i)8-s + (−2.75 − 1.58i)9-s + (1.43 − 2.81i)10-s + (−4.62 + 2.67i)11-s + (2.91 + 4.02i)12-s + (−2.61 + 2.61i)13-s + (−5.13 + 2.12i)15-s + (−0.818 − 3.91i)16-s + (−0.381 + 1.42i)17-s + (0.356 − 4.47i)18-s + (−0.130 + 0.225i)19-s + ⋯
L(s)  = 1  + (0.429 + 0.902i)2-s + (0.371 − 1.38i)3-s + (−0.630 + 0.776i)4-s + (−0.609 − 0.793i)5-s + (1.41 − 0.260i)6-s + (−0.971 − 0.235i)8-s + (−0.916 − 0.529i)9-s + (0.454 − 0.890i)10-s + (−1.39 + 0.805i)11-s + (0.841 + 1.16i)12-s + (−0.726 + 0.726i)13-s + (−1.32 + 0.549i)15-s + (−0.204 − 0.978i)16-s + (−0.0925 + 0.345i)17-s + (0.0839 − 1.05i)18-s + (−0.0298 + 0.0517i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 - 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.992 - 0.123i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.992 - 0.123i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00496660 + 0.0799093i\)
\(L(\frac12)\) \(\approx\) \(0.00496660 + 0.0799093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.607 - 1.27i)T \)
5 \( 1 + (1.36 + 1.77i)T \)
7 \( 1 \)
good3 \( 1 + (-0.643 + 2.40i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (4.62 - 2.67i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.61 - 2.61i)T - 13iT^{2} \)
17 \( 1 + (0.381 - 1.42i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.130 - 0.225i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.44 + 0.388i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.36iT - 29T^{2} \)
31 \( 1 + (-1.77 + 1.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (11.2 - 3.00i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 9.70T + 41T^{2} \)
43 \( 1 + (-6.86 - 6.86i)T + 43iT^{2} \)
47 \( 1 + (0.357 + 1.33i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.48 + 0.665i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.26 + 3.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.06 + 3.58i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.78 + 1.81i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.29iT - 71T^{2} \)
73 \( 1 + (13.1 + 3.53i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.09 + 8.83i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (5.83 + 5.83i)T + 83iT^{2} \)
89 \( 1 + (6.60 + 3.81i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-11.5 - 11.5i)T + 97iT^{2} \)
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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

−10.24787960632507387996911798175, −9.096746269512203931338151868606, −8.377570743806596140538551754748, −7.66659917687074329637029496183, −7.24942046979202276745314093694, −6.38398359667851269613058612136, −5.13659827660604326279210471562, −4.56297689560744432340993156451, −3.12815465914378516449389154874, −1.85871998885965524070153837973, 0.02861136616103758340946457656, 2.64104601033880090132806585289, 3.13990002502483441435627290574, 4.00869957551735090003683859432, 5.01430369747954014554387240218, 5.61312251399553563633695061257, 7.12844130600293498057178951372, 8.252444136020028031859844057845, 8.957969662475199286569137681770, 10.12156461785199363573552600880

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