Properties

Label 2-980-140.107-c1-0-105
Degree $2$
Conductor $980$
Sign $-0.967 + 0.251i$
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  + (1.22 − 0.702i)2-s + (0.555 − 2.07i)3-s + (1.01 − 1.72i)4-s + (−1.72 − 1.42i)5-s + (−0.773 − 2.93i)6-s + (0.0350 − 2.82i)8-s + (−1.39 − 0.806i)9-s + (−3.11 − 0.531i)10-s + (4.97 − 2.87i)11-s + (−3.01 − 3.06i)12-s + (−1.35 + 1.35i)13-s + (−3.90 + 2.79i)15-s + (−1.94 − 3.49i)16-s + (−1.16 + 4.35i)17-s + (−2.28 − 0.00942i)18-s + (2.82 − 4.90i)19-s + ⋯
L(s)  = 1  + (0.868 − 0.496i)2-s + (0.320 − 1.19i)3-s + (0.507 − 0.861i)4-s + (−0.772 − 0.635i)5-s + (−0.315 − 1.19i)6-s + (0.0123 − 0.999i)8-s + (−0.465 − 0.268i)9-s + (−0.985 − 0.168i)10-s + (1.50 − 0.866i)11-s + (−0.869 − 0.884i)12-s + (−0.374 + 0.374i)13-s + (−1.00 + 0.721i)15-s + (−0.485 − 0.874i)16-s + (−0.283 + 1.05i)17-s + (−0.537 − 0.00222i)18-s + (0.649 − 1.12i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.967 + 0.251i$
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: \((2,\ 980,\ (\ :1/2),\ -0.967 + 0.251i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.337992 - 2.64838i\)
\(L(\frac12)\) \(\approx\) \(0.337992 - 2.64838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.702i)T \)
5 \( 1 + (1.72 + 1.42i)T \)
7 \( 1 \)
good3 \( 1 + (-0.555 + 2.07i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-4.97 + 2.87i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.35 - 1.35i)T - 13iT^{2} \)
17 \( 1 + (1.16 - 4.35i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (-2.82 + 4.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.69 - 0.721i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.11iT - 29T^{2} \)
31 \( 1 + (6.52 - 3.76i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.55 - 0.683i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 3.28T + 41T^{2} \)
43 \( 1 + (-4.72 - 4.72i)T + 43iT^{2} \)
47 \( 1 + (-1.45 - 5.43i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.836 + 0.224i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.45 + 4.25i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-7.67 + 13.2i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.220 + 0.0590i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.54iT - 71T^{2} \)
73 \( 1 + (-12.0 - 3.22i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.89 + 3.27i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.12 - 8.12i)T + 83iT^{2} \)
89 \( 1 + (6.91 + 3.99i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.28 - 6.28i)T + 97iT^{2} \)
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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

−9.428352504694831505802598914184, −8.821716638199713573383282224802, −7.79635865992857633155920260860, −6.91002610159556239428790339458, −6.32896908748002575862768860872, −5.13563243278610865331297559187, −4.08916975805274616303737075012, −3.28553168537333590907217309712, −1.84357662244353450664131212353, −0.959334933726734298969103853499, 2.42894901671783964499253327086, 3.75336043447190010829301979858, 3.93545538785705176068178549587, 4.90888962930381209312446397132, 6.01316756848610596914889121002, 7.10902183038164059997908554428, 7.57024292188401684072358083835, 8.741665811939700767058212939446, 9.584839080306226793360327559604, 10.32794468695799338137296006601

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