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Properties

Label 2-980-140.139-c1-0-74
Degree $2$
Conductor $980$
Sign $-0.472 + 0.881i$
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  − 1.41i·2-s − 2.00·4-s + (0.158 − 2.23i)5-s + 2.82i·8-s + 3·9-s + (−3.15 − 0.224i)10-s + 5.99·13-s + 4.00·16-s + 6.62·17-s − 4.24i·18-s + (−0.317 + 4.46i)20-s + (−4.94 − 0.707i)25-s − 8.47i·26-s − 9.89·29-s − 5.65i·32-s + ⋯
L(s)  = 1  − 0.999i·2-s − 1.00·4-s + (0.0708 − 0.997i)5-s + 1.00i·8-s + 9-s + (−0.997 − 0.0708i)10-s + 1.66·13-s + 1.00·16-s + 1.60·17-s − 0.999i·18-s + (−0.0708 + 0.997i)20-s + (−0.989 − 0.141i)25-s − 1.66i·26-s − 1.83·29-s − 1.00i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.472 + 0.881i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (979, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.472 + 0.881i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860145 - 1.43758i\)
\(L(\frac12)\) \(\approx\) \(0.860145 - 1.43758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 + (-0.158 + 2.23i)T \)
7 \( 1 \)
good3 \( 1 - 3T^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 - 5.99T + 13T^{2} \)
17 \( 1 - 6.62T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 9.89T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 7.07iT - 37T^{2} \)
41 \( 1 + 3.56iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 14iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 13.8iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 11.6T + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 3.11iT - 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−9.636437849109940022861118187573, −9.193634819122247950585145870667, −8.239511588450579117222914364098, −7.52576931881050474204838169439, −5.93428324079188458724625952669, −5.25940457414331492534619217059, −4.08274291790926662493067706496, −3.55509950221589065541971984332, −1.80814891417734309138307363278, −0.977384854848314537471369101836, 1.38740527189310000190276454998, 3.38096788044254400675961879516, 3.96640123125089604310358061588, 5.35188472014075751979744707810, 6.12911957855876674735476975560, 6.88776476454730379408295172118, 7.64819984190347340837010196944, 8.346596803063869237013120817782, 9.509378590529418821112090061179, 10.06258485585189938401681240794

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