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Properties

Label 2-980-35.9-c1-0-12
Degree $2$
Conductor $980$
Sign $0.795 + 0.605i$
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  + (−1.01 + 0.586i)3-s + (1.93 + 1.11i)5-s + (−0.811 + 1.40i)9-s + (−3.31 − 5.73i)11-s − 5.64i·13-s − 2.62·15-s + (6.92 − 3.99i)17-s + (2.5 + 4.33i)25-s − 5.42i·27-s + 0.623·29-s + (6.72 + 3.88i)33-s + (3.31 + 5.73i)39-s + (−3.14 + 1.81i)45-s + (10.7 + 6.23i)47-s + (−4.68 + 8.12i)51-s + ⋯
L(s)  = 1  + (−0.586 + 0.338i)3-s + (0.866 + 0.499i)5-s + (−0.270 + 0.468i)9-s + (−0.998 − 1.72i)11-s − 1.56i·13-s − 0.677·15-s + (1.67 − 0.969i)17-s + (0.5 + 0.866i)25-s − 1.04i·27-s + 0.115·29-s + (1.17 + 0.676i)33-s + (0.530 + 0.918i)39-s + (−0.468 + 0.270i)45-s + (1.57 + 0.909i)47-s + (−0.656 + 1.13i)51-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.795 + 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.795 + 0.605i$
Analytic conductor: \(7.82533\)
Root analytic conductor: \(2.79738\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :1/2),\ 0.795 + 0.605i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24406 - 0.419428i\)
\(L(\frac12)\) \(\approx\) \(1.24406 - 0.419428i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.93 - 1.11i)T \)
7 \( 1 \)
good3 \( 1 + (1.01 - 0.586i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (3.31 + 5.73i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.64iT - 13T^{2} \)
17 \( 1 + (-6.92 + 3.99i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 0.623T + 29T^{2} \)
31 \( 1 + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + (-10.7 - 6.23i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-11.6 + 6.70i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.93 + 13.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 8.94iT - 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.6iT - 97T^{2} \)
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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

−10.35906361304640651247160017753, −9.220704983233821272447726550507, −8.090745312740185939693000995566, −7.57677109018905975823421938853, −6.06519451603816321292418613570, −5.59667336549362771466497033222, −5.11115358690030335998791603661, −3.23966693218574834391675063688, −2.72964088163183375706655033515, −0.70519999087873212843954240412, 1.36003256903670242506017576069, 2.33989717477191444408424416751, 3.98921730288870766695410234553, 5.05363281352300428312117309335, 5.73458807817803587971773779737, 6.64856686172375747197084830516, 7.39498397936190863763913642402, 8.481533857469857500592091547375, 9.465720218373956946031680862349, 9.959058882058710604583493705402

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