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Properties

Label 2-980-7.2-c1-0-8
Degree $2$
Conductor $980$
Sign $0.701 + 0.712i$
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  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (1 − 1.73i)9-s + (−1.5 − 2.59i)11-s + 13-s + 0.999·15-s + (−1.5 − 2.59i)17-s + (1 − 1.73i)19-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s + 5·27-s − 9·29-s + (4 + 6.92i)31-s + (1.5 − 2.59i)33-s + (5 − 8.66i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (0.333 − 0.577i)9-s + (−0.452 − 0.783i)11-s + 0.277·13-s + 0.258·15-s + (−0.363 − 0.630i)17-s + (0.229 − 0.397i)19-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s + 0.962·27-s − 1.67·29-s + (0.718 + 1.24i)31-s + (0.261 − 0.452i)33-s + (0.821 − 1.42i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61000 - 0.674727i\)
\(L(\frac12)\) \(\approx\) \(1.61000 - 0.674727i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - T + 13T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (1.5 - 2.59i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-7 - 12.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.5 - 4.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 17T + 97T^{2} \)
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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

−9.743090891685292307546137647562, −9.062667533515878684049021687528, −8.517263805985110385723093610809, −7.38095680544537130193362580007, −6.46679365889822719237625963845, −5.46890453147666287549010541676, −4.57426435890614877901453642291, −3.58188869311409331164942614819, −2.55001106960615566087246522037, −0.822239012411896174393442058562, 1.59004638298352418964094952922, 2.49136670286715251620276324972, 3.76178797567082590955893838189, 4.88775740636797138867394016278, 5.88003643847922287329289675124, 6.86955817536010857967183384958, 7.62767629169851764066551470561, 8.194868196530131526967800745369, 9.410702100232467724326086596023, 10.03077824192863008076771116252

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