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Properties

Label 2-980-140.107-c1-0-22
Degree $2$
Conductor $980$
Sign $-0.451 - 0.892i$
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  + (1.02 − 0.970i)2-s + (−0.517 + 1.93i)3-s + (0.115 − 1.99i)4-s + (0.653 + 2.13i)5-s + (1.34 + 2.49i)6-s + (−1.81 − 2.16i)8-s + (−0.870 − 0.502i)9-s + (2.74 + 1.56i)10-s + (−4.92 + 2.84i)11-s + (3.79 + 1.25i)12-s + (−4.82 + 4.82i)13-s + (−4.47 + 0.155i)15-s + (−3.97 − 0.463i)16-s + (0.979 − 3.65i)17-s + (−1.38 + 0.327i)18-s + (0.564 − 0.977i)19-s + ⋯
L(s)  = 1  + (0.727 − 0.686i)2-s + (−0.299 + 1.11i)3-s + (0.0579 − 0.998i)4-s + (0.292 + 0.956i)5-s + (0.548 + 1.01i)6-s + (−0.642 − 0.765i)8-s + (−0.290 − 0.167i)9-s + (0.868 + 0.494i)10-s + (−1.48 + 0.856i)11-s + (1.09 + 0.363i)12-s + (−1.33 + 1.33i)13-s + (−1.15 + 0.0401i)15-s + (−0.993 − 0.115i)16-s + (0.237 − 0.886i)17-s + (−0.325 + 0.0772i)18-s + (0.129 − 0.224i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.451 - 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.451 - 0.892i$
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: \((2,\ 980,\ (\ :1/2),\ -0.451 - 0.892i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.699074 + 1.13656i\)
\(L(\frac12)\) \(\approx\) \(0.699074 + 1.13656i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.02 + 0.970i)T \)
5 \( 1 + (-0.653 - 2.13i)T \)
7 \( 1 \)
good3 \( 1 + (0.517 - 1.93i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (4.92 - 2.84i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.82 - 4.82i)T - 13iT^{2} \)
17 \( 1 + (-0.979 + 3.65i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.564 + 0.977i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4.70 + 1.26i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 0.781iT - 29T^{2} \)
31 \( 1 + (3.20 - 1.84i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.32 + 0.889i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 0.451T + 41T^{2} \)
43 \( 1 + (0.613 + 0.613i)T + 43iT^{2} \)
47 \( 1 + (-1.10 - 4.14i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.03 + 0.544i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.63 - 8.03i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.759 - 1.31i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-13.1 - 3.53i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (-1.66 - 0.445i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.73 - 4.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-0.968 - 0.968i)T + 83iT^{2} \)
89 \( 1 + (8.16 + 4.71i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.77 - 6.77i)T + 97iT^{2} \)
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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

−10.25387685715272608189901762919, −9.805430550011990423639494490262, −9.247168342334730183511200268334, −7.36448868481133367545655835759, −6.89566536911466071959304946941, −5.46882872695057910866104165262, −4.93193517616907740616632529657, −4.19016869671033955357273309699, −2.89517231148849223461142107518, −2.21473286189472305932663795410, 0.46175636440420256575565203673, 2.20614063930517380206871810357, 3.34528596857595692507204736814, 4.89923595170796239125643146299, 5.46585111803510875893323161647, 6.09814695171933795800734810291, 7.27948189833807458709974706349, 7.912460005093465188053374078166, 8.356992996863981840515146387757, 9.628884123669378019884156454768

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