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Properties

Label 2-980-140.39-c0-0-1
Degree $2$
Conductor $980$
Sign $0.944 - 0.328i$
Analytic cond. $0.489083$
Root an. cond. $0.699345$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (0.965 − 0.258i)10-s − 1.41i·13-s + (−0.5 − 0.866i)16-s + (1.22 + 0.707i)17-s + (−0.866 − 0.499i)18-s + (−0.707 + 0.707i)20-s + (0.866 + 0.499i)25-s + (0.707 + 1.22i)26-s + (0.866 + 0.499i)32-s − 1.41·34-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.965 − 0.258i)5-s + 0.999i·8-s + (0.5 + 0.866i)9-s + (0.965 − 0.258i)10-s − 1.41i·13-s + (−0.5 − 0.866i)16-s + (1.22 + 0.707i)17-s + (−0.866 − 0.499i)18-s + (−0.707 + 0.707i)20-s + (0.866 + 0.499i)25-s + (0.707 + 1.22i)26-s + (0.866 + 0.499i)32-s − 1.41·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(980\)    =    \(2^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.944 - 0.328i$
Analytic conductor: \(0.489083\)
Root analytic conductor: \(0.699345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{980} (459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 980,\ (\ :0),\ 0.944 - 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6215799039\)
\(L(\frac12)\) \(\approx\) \(0.6215799039\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.41iT - T^{2} \)
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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

−10.31039097304477237609774471422, −9.350958932551236198169326493779, −8.299000091668864983571778507251, −7.77847901556412539578274787496, −7.35254388748331778906011670772, −5.97175471216079513764284289979, −5.24837717668583242438779797325, −4.11615482734074465580480162201, −2.72931873858675939607097594529, −1.09103090404951139878806406777, 1.12503455848332174329274823098, 2.73960846638942985615717002133, 3.74268858308021409640504315515, 4.48927676494909329218206359569, 6.25535783319026214325310758899, 7.06652642881219069541522391606, 7.67021436107121208520001598496, 8.583758314404481453244171368113, 9.493600949540969785583798795577, 9.922552197709276684957316387725

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