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Properties

Label 2-980-140.107-c1-0-100
Degree $2$
Conductor $980$
Sign $-0.983 + 0.182i$
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  + (−1.41 − 0.0109i)2-s + (0.646 − 2.41i)3-s + (1.99 + 0.0309i)4-s + (0.0689 − 2.23i)5-s + (−0.940 + 3.40i)6-s + (−2.82 − 0.0656i)8-s + (−2.80 − 1.61i)9-s + (−0.121 + 3.15i)10-s + (2.41 − 1.39i)11-s + (1.36 − 4.80i)12-s + (0.850 − 0.850i)13-s + (−5.34 − 1.61i)15-s + (3.99 + 0.123i)16-s + (−0.545 + 2.03i)17-s + (3.94 + 2.32i)18-s + (2.35 − 4.08i)19-s + ⋯
L(s)  = 1  + (−0.999 − 0.00774i)2-s + (0.373 − 1.39i)3-s + (0.999 + 0.0154i)4-s + (0.0308 − 0.999i)5-s + (−0.383 + 1.38i)6-s + (−0.999 − 0.0232i)8-s + (−0.934 − 0.539i)9-s + (−0.0385 + 0.999i)10-s + (0.726 − 0.419i)11-s + (0.394 − 1.38i)12-s + (0.235 − 0.235i)13-s + (−1.38 − 0.415i)15-s + (0.999 + 0.0309i)16-s + (−0.132 + 0.493i)17-s + (0.930 + 0.546i)18-s + (0.541 − 0.937i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.182i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.983 + 0.182i$
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: \((2,\ 980,\ (\ :1/2),\ -0.983 + 0.182i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0968037 - 1.05471i\)
\(L(\frac12)\) \(\approx\) \(0.0968037 - 1.05471i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0109i)T \)
5 \( 1 + (-0.0689 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 + (-0.646 + 2.41i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-2.41 + 1.39i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.850 + 0.850i)T - 13iT^{2} \)
17 \( 1 + (0.545 - 2.03i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.35 + 4.08i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.67 - 2.05i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.15iT - 29T^{2} \)
31 \( 1 + (-8.84 + 5.10i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (7.20 - 1.93i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.44T + 41T^{2} \)
43 \( 1 + (-4.72 - 4.72i)T + 43iT^{2} \)
47 \( 1 + (-0.509 - 1.90i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.56 - 0.954i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.07 - 8.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.80 - 4.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.28 - 0.881i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 4.45iT - 71T^{2} \)
73 \( 1 + (-3.56 - 0.955i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.46 - 7.74i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.940 - 0.940i)T + 83iT^{2} \)
89 \( 1 + (-4.46 - 2.57i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.28 + 1.28i)T + 97iT^{2} \)
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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

−9.444222204557671936496080737036, −8.387979376180257063032262919522, −8.246709098433644085024970893981, −7.31525241611973327671299031804, −6.37117162633009802068472572748, −5.76567328733737756492345796722, −4.08189060600652987718953462322, −2.60198933645531652165217025608, −1.57667293103336659011424328402, −0.65927586119598188022680519383, 1.89508394150612160520911524687, 3.20873151704678089349044542291, 3.82286784211109630501255214621, 5.18245253165751874577321583148, 6.40297499083035340413329133376, 7.05423067538042196921535447998, 8.162970967487014125920388897890, 8.917856166216757459186204462625, 9.695115489819529565023908129324, 10.30907197440110515614759002324

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