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Properties

Label 2-980-140.107-c1-0-82
Degree $2$
Conductor $980$
Sign $0.723 + 0.690i$
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  + (0.764 + 1.18i)2-s + (0.638 − 2.38i)3-s + (−0.831 + 1.81i)4-s + (−2.14 + 0.631i)5-s + (3.32 − 1.06i)6-s + (−2.79 + 0.401i)8-s + (−2.68 − 1.54i)9-s + (−2.39 − 2.06i)10-s + (4.09 − 2.36i)11-s + (3.80 + 3.14i)12-s + (−0.0592 + 0.0592i)13-s + (0.135 + 5.51i)15-s + (−2.61 − 3.02i)16-s + (1.27 − 4.77i)17-s + (−0.207 − 4.37i)18-s + (1.31 − 2.27i)19-s + ⋯
L(s)  = 1  + (0.540 + 0.841i)2-s + (0.368 − 1.37i)3-s + (−0.415 + 0.909i)4-s + (−0.959 + 0.282i)5-s + (1.35 − 0.433i)6-s + (−0.989 + 0.142i)8-s + (−0.893 − 0.515i)9-s + (−0.756 − 0.654i)10-s + (1.23 − 0.713i)11-s + (1.09 + 0.907i)12-s + (−0.0164 + 0.0164i)13-s + (0.0349 + 1.42i)15-s + (−0.654 − 0.755i)16-s + (0.310 − 1.15i)17-s + (−0.0490 − 1.03i)18-s + (0.301 − 0.522i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.723 + 0.690i$
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: \((2,\ 980,\ (\ :1/2),\ 0.723 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.69695 - 0.680351i\)
\(L(\frac12)\) \(\approx\) \(1.69695 - 0.680351i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.764 - 1.18i)T \)
5 \( 1 + (2.14 - 0.631i)T \)
7 \( 1 \)
good3 \( 1 + (-0.638 + 2.38i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-4.09 + 2.36i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.0592 - 0.0592i)T - 13iT^{2} \)
17 \( 1 + (-1.27 + 4.77i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.31 + 2.27i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.09 + 0.292i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 7.27iT - 29T^{2} \)
31 \( 1 + (4.01 - 2.31i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.22 + 0.596i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 5.71T + 41T^{2} \)
43 \( 1 + (1.57 + 1.57i)T + 43iT^{2} \)
47 \( 1 + (-0.716 - 2.67i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (9.12 + 2.44i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-1.67 - 2.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.978 + 1.69i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.491 - 0.131i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 14.4iT - 71T^{2} \)
73 \( 1 + (9.26 + 2.48i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-3.05 + 5.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.62 - 5.62i)T + 83iT^{2} \)
89 \( 1 + (-14.4 - 8.34i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.81 + 5.81i)T + 97iT^{2} \)
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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

−9.459140008979056676654746240054, −8.690071275996533689365808138271, −7.914367600797411491028895180719, −7.28403353180398815360425083724, −6.72964145604353105361118710752, −5.91644747822686293995387497219, −4.60602539155587123508208048525, −3.55918642021376437256509394625, −2.66051927037052477928947896934, −0.74858965915043058834548925503, 1.50594321282867162266931022045, 3.24496623539383275806124298194, 3.85030196602441467047800825524, 4.43638746080348604642911484990, 5.28817366333691118649779487219, 6.50475215203152598329708395535, 7.80803065982434432400005642258, 8.969911548130567842575408030937, 9.275600862132040671025994647486, 10.23471533213843135505677707411

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