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Properties

Label 2-880-11.3-c1-0-13
Degree $2$
Conductor $880$
Sign $0.966 - 0.255i$
Analytic cond. $7.02683$
Root an. cond. $2.65081$
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  + (0.400 + 1.23i)3-s + (−0.809 − 0.587i)5-s + (0.0703 − 0.216i)7-s + (1.07 − 0.777i)9-s + (3.12 + 1.12i)11-s + (2.70 − 1.96i)13-s + (0.400 − 1.23i)15-s + (−4.89 − 3.55i)17-s + (0.686 + 2.11i)19-s + 0.294·21-s + 3.47·23-s + (0.309 + 0.951i)25-s + (4.52 + 3.29i)27-s + (0.00951 − 0.0292i)29-s + (1.79 − 1.30i)31-s + ⋯
L(s)  = 1  + (0.231 + 0.711i)3-s + (−0.361 − 0.262i)5-s + (0.0266 − 0.0818i)7-s + (0.356 − 0.259i)9-s + (0.940 + 0.339i)11-s + (0.750 − 0.545i)13-s + (0.103 − 0.317i)15-s + (−1.18 − 0.862i)17-s + (0.157 + 0.484i)19-s + 0.0643·21-s + 0.725·23-s + (0.0618 + 0.190i)25-s + (0.871 + 0.633i)27-s + (0.00176 − 0.00543i)29-s + (0.322 − 0.233i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.966 - 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(880\)    =    \(2^{4} \cdot 5 \cdot 11\)
Sign: $0.966 - 0.255i$
Analytic conductor: \(7.02683\)
Root analytic conductor: \(2.65081\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{880} (641, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 880,\ (\ :1/2),\ 0.966 - 0.255i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77019 + 0.230224i\)
\(L(\frac12)\) \(\approx\) \(1.77019 + 0.230224i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (-3.12 - 1.12i)T \)
good3 \( 1 + (-0.400 - 1.23i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.0703 + 0.216i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-2.70 + 1.96i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (4.89 + 3.55i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.686 - 2.11i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 3.47T + 23T^{2} \)
29 \( 1 + (-0.00951 + 0.0292i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-1.79 + 1.30i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.70 + 8.31i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.16 - 9.73i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.74T + 43T^{2} \)
47 \( 1 + (-1.64 - 5.05i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.25 + 5.99i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (2.57 - 7.93i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-1.81 - 1.31i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 3.47T + 67T^{2} \)
71 \( 1 + (12.1 + 8.85i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (1.13 - 3.48i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (9.40 - 6.83i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.23 - 0.895i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + (-6.74 + 4.89i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.09109557346846065601896581889, −9.185496224210368642816691598543, −8.843194789965452081303558628810, −7.62399523623551963329232377312, −6.81138175953625877139144421129, −5.77234093389210678557525385360, −4.47663894912375436577711937731, −4.05866887802107880063012190785, −2.87275156309797732054835854435, −1.11730891898724885984994797261, 1.21730162971412849370097203410, 2.39917781499020534785091131654, 3.75586863735114588304091502852, 4.57513927904809729279464545586, 6.05352225239684001050788471452, 6.78402492756081025289814835324, 7.39160142642993087185389114122, 8.645825648539020383174156213740, 8.843772481154164465787582030486, 10.22041948179272740801329054933

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