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Properties

Label 2-980-140.107-c1-0-14
Degree $2$
Conductor $980$
Sign $-0.0364 + 0.999i$
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  + (0.491 + 1.32i)2-s + (−0.368 + 1.37i)3-s + (−1.51 + 1.30i)4-s + (−1.35 + 1.78i)5-s + (−2.00 + 0.186i)6-s + (−2.47 − 1.37i)8-s + (0.838 + 0.483i)9-s + (−3.02 − 0.917i)10-s + (−0.404 + 0.233i)11-s + (−1.23 − 2.57i)12-s + (−2.66 + 2.66i)13-s + (−1.95 − 2.51i)15-s + (0.607 − 3.95i)16-s + (1.19 − 4.45i)17-s + (−0.230 + 1.34i)18-s + (−3.44 + 5.96i)19-s + ⋯
L(s)  = 1  + (0.347 + 0.937i)2-s + (−0.213 + 0.794i)3-s + (−0.758 + 0.651i)4-s + (−0.604 + 0.796i)5-s + (−0.819 + 0.0762i)6-s + (−0.874 − 0.485i)8-s + (0.279 + 0.161i)9-s + (−0.957 − 0.290i)10-s + (−0.121 + 0.0704i)11-s + (−0.356 − 0.742i)12-s + (−0.738 + 0.738i)13-s + (−0.504 − 0.650i)15-s + (0.151 − 0.988i)16-s + (0.289 − 1.08i)17-s + (−0.0542 + 0.318i)18-s + (−0.790 + 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0364 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0364 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.0364 + 0.999i$
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: \((2,\ 980,\ (\ :1/2),\ -0.0364 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.510215 - 0.529174i\)
\(L(\frac12)\) \(\approx\) \(0.510215 - 0.529174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.491 - 1.32i)T \)
5 \( 1 + (1.35 - 1.78i)T \)
7 \( 1 \)
good3 \( 1 + (0.368 - 1.37i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (0.404 - 0.233i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.66 - 2.66i)T - 13iT^{2} \)
17 \( 1 + (-1.19 + 4.45i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.44 - 5.96i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.04 - 0.816i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 2.62iT - 29T^{2} \)
31 \( 1 + (-2.86 + 1.65i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-9.91 + 2.65i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.58T + 41T^{2} \)
43 \( 1 + (1.82 + 1.82i)T + 43iT^{2} \)
47 \( 1 + (0.865 + 3.22i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-10.5 - 2.82i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-5.72 - 9.92i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.20 - 2.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (14.7 + 3.95i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 9.62iT - 71T^{2} \)
73 \( 1 + (3.12 + 0.838i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.45 - 2.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (8.69 + 8.69i)T + 83iT^{2} \)
89 \( 1 + (1.64 + 0.947i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.04 + 6.04i)T + 97iT^{2} \)
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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

−10.26590073756912706855340917656, −9.937648904915936086909354895374, −8.901388376986325604708811099777, −7.77432608453081568263303754821, −7.34929404274242726190040304242, −6.36379165840727690780262135011, −5.44510722163102474006234136079, −4.32428412133196273664573574049, −3.98095972487124821696104314594, −2.60774925606276236590818658915, 0.32161546238901078093186330234, 1.46186580901784395494968303651, 2.73358699430284980728428395028, 4.02341573720723408496525907857, 4.76419981782545058688926959194, 5.76915311043247414748417703698, 6.75512502685129014776009507700, 7.901607109157474937434293841352, 8.511192877504031903460730560011, 9.530382994498828021825393102383

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