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Properties

Label 2-980-140.107-c1-0-16
Degree $2$
Conductor $980$
Sign $-0.999 + 0.0395i$
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  + (0.0366 + 1.41i)2-s + (−0.107 + 0.402i)3-s + (−1.99 + 0.103i)4-s + (−1.27 − 1.83i)5-s + (−0.572 − 0.137i)6-s + (−0.219 − 2.81i)8-s + (2.44 + 1.41i)9-s + (2.54 − 1.87i)10-s + (−0.725 + 0.418i)11-s + (0.173 − 0.814i)12-s + (1.16 − 1.16i)13-s + (0.875 − 0.316i)15-s + (3.97 − 0.414i)16-s + (−1.32 + 4.94i)17-s + (−1.90 + 3.51i)18-s + (−2.91 + 5.05i)19-s + ⋯
L(s)  = 1  + (0.0259 + 0.999i)2-s + (−0.0622 + 0.232i)3-s + (−0.998 + 0.0518i)4-s + (−0.571 − 0.820i)5-s + (−0.233 − 0.0561i)6-s + (−0.0777 − 0.996i)8-s + (0.815 + 0.471i)9-s + (0.805 − 0.592i)10-s + (−0.218 + 0.126i)11-s + (0.0500 − 0.235i)12-s + (0.322 − 0.322i)13-s + (0.226 − 0.0815i)15-s + (0.994 − 0.103i)16-s + (−0.321 + 1.20i)17-s + (−0.449 + 0.827i)18-s + (−0.669 + 1.15i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0395i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.999 + 0.0395i$
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: \((2,\ 980,\ (\ :1/2),\ -0.999 + 0.0395i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0138881 - 0.701626i\)
\(L(\frac12)\) \(\approx\) \(0.0138881 - 0.701626i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0366 - 1.41i)T \)
5 \( 1 + (1.27 + 1.83i)T \)
7 \( 1 \)
good3 \( 1 + (0.107 - 0.402i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.725 - 0.418i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.16 + 1.16i)T - 13iT^{2} \)
17 \( 1 + (1.32 - 4.94i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (2.91 - 5.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.34 - 0.896i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.00iT - 29T^{2} \)
31 \( 1 + (7.03 - 4.06i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-0.711 + 0.190i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.0958T + 41T^{2} \)
43 \( 1 + (-4.87 - 4.87i)T + 43iT^{2} \)
47 \( 1 + (1.41 + 5.28i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (12.8 + 3.43i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.46 - 7.74i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.919 - 1.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.515 + 0.138i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 13.6iT - 71T^{2} \)
73 \( 1 + (-5.78 - 1.55i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.30 - 9.19i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.36 - 4.36i)T + 83iT^{2} \)
89 \( 1 + (2.50 + 1.44i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.24 + 4.24i)T + 97iT^{2} \)
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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

−10.20244576433382643031860011381, −9.491658877585423878791217148128, −8.360869791194107953892134434467, −8.088624916903804454854822074708, −7.17199333081187691390051451056, −6.05390123442713249132961678538, −5.30660445864891457531044553557, −4.23387088080026347484091626088, −3.83378564541456387338318041835, −1.56457581960555800867746712552, 0.33469488041576019592667794613, 2.00651895528529834598202738420, 3.07896598157908455976438589188, 4.02556100717532855196312025544, 4.84408545222819887069500063817, 6.23330925756039670905469274728, 7.10914826054109102627948514941, 7.905107101426774704146528725990, 9.063506131742378682160188102251, 9.595230182199520140535639328932

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