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Properties

Label 2-980-35.13-c1-0-3
Degree $2$
Conductor $980$
Sign $-0.262 - 0.964i$
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  + (−1.42 + 1.42i)3-s + (−2.11 − 0.724i)5-s − 1.06i·9-s + 5.50·11-s + (1.36 − 1.36i)13-s + (4.04 − 1.98i)15-s + (0.849 + 0.849i)17-s − 0.519·19-s + (−4.91 − 4.91i)23-s + (3.95 + 3.06i)25-s + (−2.75 − 2.75i)27-s + 9.66i·29-s + 3.60i·31-s + (−7.85 + 7.85i)33-s + (−1.27 + 1.27i)37-s + ⋯
L(s)  = 1  + (−0.823 + 0.823i)3-s + (−0.946 − 0.323i)5-s − 0.354i·9-s + 1.66·11-s + (0.379 − 0.379i)13-s + (1.04 − 0.512i)15-s + (0.205 + 0.205i)17-s − 0.119·19-s + (−1.02 − 1.02i)23-s + (0.790 + 0.612i)25-s + (−0.530 − 0.530i)27-s + 1.79i·29-s + 0.646i·31-s + (−1.36 + 1.36i)33-s + (−0.210 + 0.210i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.262 - 0.964i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ -0.262 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.521354 + 0.682355i\)
\(L(\frac12)\) \(\approx\) \(0.521354 + 0.682355i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (2.11 + 0.724i)T \)
7 \( 1 \)
good3 \( 1 + (1.42 - 1.42i)T - 3iT^{2} \)
11 \( 1 - 5.50T + 11T^{2} \)
13 \( 1 + (-1.36 + 1.36i)T - 13iT^{2} \)
17 \( 1 + (-0.849 - 0.849i)T + 17iT^{2} \)
19 \( 1 + 0.519T + 19T^{2} \)
23 \( 1 + (4.91 + 4.91i)T + 23iT^{2} \)
29 \( 1 - 9.66iT - 29T^{2} \)
31 \( 1 - 3.60iT - 31T^{2} \)
37 \( 1 + (1.27 - 1.27i)T - 37iT^{2} \)
41 \( 1 - 9.63iT - 41T^{2} \)
43 \( 1 + (-5.88 - 5.88i)T + 43iT^{2} \)
47 \( 1 + (-3.91 - 3.91i)T + 47iT^{2} \)
53 \( 1 + (6.06 + 6.06i)T + 53iT^{2} \)
59 \( 1 + 4.02T + 59T^{2} \)
61 \( 1 - 12.0iT - 61T^{2} \)
67 \( 1 + (10.1 - 10.1i)T - 67iT^{2} \)
71 \( 1 + 1.96T + 71T^{2} \)
73 \( 1 + (-10.1 + 10.1i)T - 73iT^{2} \)
79 \( 1 + 9.86iT - 79T^{2} \)
83 \( 1 + (5.05 - 5.05i)T - 83iT^{2} \)
89 \( 1 - 4.50T + 89T^{2} \)
97 \( 1 + (-10.3 - 10.3i)T + 97iT^{2} \)
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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

−10.48369516588753548309411307405, −9.407745864410117187894919163847, −8.685358718480950727745139241087, −7.82233631642868217695237323006, −6.68614590770836310029261562760, −5.93061476970548194948418761684, −4.77639324748937834104178628185, −4.21159221821758405408268442199, −3.30942395806500690640253757322, −1.22219381293070847417423454872, 0.52426009799280054126164288562, 1.86457297625212403198618803460, 3.64760713269628767178560950998, 4.21803327029112316266952461930, 5.77169021248879438090418241572, 6.36856872757139123411020508530, 7.17143387064232417787262741502, 7.81124137888838073294448954871, 8.917976584625241327525823536572, 9.706228873984825366796679903621

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