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Properties

Label 2-980-140.123-c1-0-5
Degree $2$
Conductor $980$
Sign $0.0134 + 0.999i$
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  + (0.604 + 1.27i)2-s + (0.844 + 3.15i)3-s + (−1.26 + 1.54i)4-s + (−1.79 − 1.33i)5-s + (−3.52 + 2.98i)6-s + (−2.74 − 0.689i)8-s + (−6.62 + 3.82i)9-s + (0.624 − 3.10i)10-s + (−1.96 − 1.13i)11-s + (−5.94 − 2.69i)12-s + (1.38 + 1.38i)13-s + (2.69 − 6.78i)15-s + (−0.775 − 3.92i)16-s + (0.0499 + 0.186i)17-s + (−8.89 − 6.15i)18-s + (3.45 + 5.98i)19-s + ⋯
L(s)  = 1  + (0.427 + 0.904i)2-s + (0.487 + 1.81i)3-s + (−0.634 + 0.772i)4-s + (−0.801 − 0.597i)5-s + (−1.43 + 1.21i)6-s + (−0.969 − 0.243i)8-s + (−2.20 + 1.27i)9-s + (0.197 − 0.980i)10-s + (−0.593 − 0.342i)11-s + (−1.71 − 0.778i)12-s + (0.383 + 0.383i)13-s + (0.696 − 1.75i)15-s + (−0.193 − 0.981i)16-s + (0.0121 + 0.0451i)17-s + (−2.09 − 1.45i)18-s + (0.792 + 1.37i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0134 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0134 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.0134 + 0.999i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.0134 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679484 - 0.670403i\)
\(L(\frac12)\) \(\approx\) \(0.679484 - 0.670403i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.604 - 1.27i)T \)
5 \( 1 + (1.79 + 1.33i)T \)
7 \( 1 \)
good3 \( 1 + (-0.844 - 3.15i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.96 + 1.13i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.38 - 1.38i)T + 13iT^{2} \)
17 \( 1 + (-0.0499 - 0.186i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-3.45 - 5.98i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.23 + 0.866i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + 7.33iT - 29T^{2} \)
31 \( 1 + (-0.430 - 0.248i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (3.37 + 0.904i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.22T + 41T^{2} \)
43 \( 1 + (2.91 - 2.91i)T - 43iT^{2} \)
47 \( 1 + (-0.645 + 2.40i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.98 - 1.87i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.61 - 4.52i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.00 - 8.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (12.0 - 3.21i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 3.60iT - 71T^{2} \)
73 \( 1 + (12.6 - 3.38i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (-5.66 - 9.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.591 - 0.591i)T - 83iT^{2} \)
89 \( 1 + (3.45 - 1.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.09 - 1.09i)T - 97iT^{2} \)
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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

−10.36083258518905835173100188248, −9.634940186825094571767712559371, −8.812564864307479162600329930521, −8.206643222819658670993523655631, −7.62825658349397665832690122735, −5.95640608059892687475044447794, −5.32144328079970953224649648028, −4.33403183763944902983486059404, −3.89603894392084962993998478234, −2.98062853792938768110717225823, 0.35358893876480299876672390923, 1.71390309052939050977138259148, 2.84628671728495739761212721107, 3.36249491526999453495217016559, 4.91215224170800187082074617035, 6.05601524718866711485939918563, 6.94402107117828579395664128910, 7.63631636554058222270904499904, 8.440778969431994352452953698159, 9.264550238492010029655755207966

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