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Properties

Label 2-980-140.67-c1-0-81
Degree $2$
Conductor $980$
Sign $0.740 + 0.671i$
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  + (−1.17 − 0.788i)2-s + (2.86 − 0.767i)3-s + (0.756 + 1.85i)4-s + (1.62 + 1.53i)5-s + (−3.96 − 1.35i)6-s + (0.572 − 2.76i)8-s + (5.01 − 2.89i)9-s + (−0.696 − 3.08i)10-s + (0.186 + 0.107i)11-s + (3.58 + 4.72i)12-s + (4.29 − 4.29i)13-s + (5.83 + 3.15i)15-s + (−2.85 + 2.80i)16-s + (−2.74 + 0.735i)17-s + (−8.16 − 0.555i)18-s + (−0.438 − 0.760i)19-s + ⋯
L(s)  = 1  + (−0.830 − 0.557i)2-s + (1.65 − 0.442i)3-s + (0.378 + 0.925i)4-s + (0.726 + 0.686i)5-s + (−1.61 − 0.554i)6-s + (0.202 − 0.979i)8-s + (1.67 − 0.964i)9-s + (−0.220 − 0.975i)10-s + (0.0561 + 0.0324i)11-s + (1.03 + 1.36i)12-s + (1.19 − 1.19i)13-s + (1.50 + 0.813i)15-s + (−0.714 + 0.700i)16-s + (−0.665 + 0.178i)17-s + (−1.92 − 0.130i)18-s + (−0.100 − 0.174i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.09702 - 0.808975i\)
\(L(\frac12)\) \(\approx\) \(2.09702 - 0.808975i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.17 + 0.788i)T \)
5 \( 1 + (-1.62 - 1.53i)T \)
7 \( 1 \)
good3 \( 1 + (-2.86 + 0.767i)T + (2.59 - 1.5i)T^{2} \)
11 \( 1 + (-0.186 - 0.107i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.29 + 4.29i)T - 13iT^{2} \)
17 \( 1 + (2.74 - 0.735i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.438 + 0.760i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.0330 - 0.123i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 - 4.03iT - 29T^{2} \)
31 \( 1 + (7.44 + 4.30i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.473 - 1.76i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.91T + 41T^{2} \)
43 \( 1 + (2.06 + 2.06i)T + 43iT^{2} \)
47 \( 1 + (-6.67 - 1.78i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.01 + 3.78i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.24 - 9.08i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.16 - 7.21i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.203 + 0.759i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 1.75iT - 71T^{2} \)
73 \( 1 + (-1.53 - 5.71i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (6.20 + 10.7i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.30 - 5.30i)T + 83iT^{2} \)
89 \( 1 + (11.0 - 6.37i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.58 + 2.58i)T + 97iT^{2} \)
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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

−9.739172448005963589103604779056, −8.966148395639289613349518014970, −8.493273781413045967650775563075, −7.58034233727153531275340691406, −6.96069203948760006142179225642, −5.89770775637463863248490339228, −3.91330195988776717211123055116, −3.15086354489501380933176339543, −2.37540643763601192501835825456, −1.40043702259333784072636194408, 1.54803117126042139484028623213, 2.32403224571948336530199652363, 3.82354876236491223380328302510, 4.78432588014617398969895483840, 6.00543311737846258444148307985, 6.91268570908037128484196147978, 7.945733412035134649660662618638, 8.700829641682839066250592246367, 9.093663440367037080795371788103, 9.598452687983291589986826611860

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