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Properties

Label 2-640-80.29-c1-0-15
Degree $2$
Conductor $640$
Sign $0.887 + 0.460i$
Analytic cond. $5.11042$
Root an. cond. $2.26062$
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  + (1.86 − 1.86i)3-s + (1.17 + 1.90i)5-s + 3.61·7-s − 3.92i·9-s + (0.0947 − 0.0947i)11-s + (−2.59 + 2.59i)13-s + (5.72 + 1.36i)15-s − 1.89i·17-s + (2.16 + 2.16i)19-s + (6.72 − 6.72i)21-s − 5.08·23-s + (−2.25 + 4.46i)25-s + (−1.71 − 1.71i)27-s + (−1.25 − 1.25i)29-s − 1.27·31-s + ⋯
L(s)  = 1  + (1.07 − 1.07i)3-s + (0.524 + 0.851i)5-s + 1.36·7-s − 1.30i·9-s + (0.0285 − 0.0285i)11-s + (−0.719 + 0.719i)13-s + (1.47 + 0.351i)15-s − 0.460i·17-s + (0.496 + 0.496i)19-s + (1.46 − 1.46i)21-s − 1.05·23-s + (−0.450 + 0.892i)25-s + (−0.329 − 0.329i)27-s + (−0.233 − 0.233i)29-s − 0.228·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 640 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.887 + 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(640\)    =    \(2^{7} \cdot 5\)
Sign: $0.887 + 0.460i$
Analytic conductor: \(5.11042\)
Root analytic conductor: \(2.26062\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{640} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 640,\ (\ :1/2),\ 0.887 + 0.460i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43694 - 0.594623i\)
\(L(\frac12)\) \(\approx\) \(2.43694 - 0.594623i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.17 - 1.90i)T \)
good3 \( 1 + (-1.86 + 1.86i)T - 3iT^{2} \)
7 \( 1 - 3.61T + 7T^{2} \)
11 \( 1 + (-0.0947 + 0.0947i)T - 11iT^{2} \)
13 \( 1 + (2.59 - 2.59i)T - 13iT^{2} \)
17 \( 1 + 1.89iT - 17T^{2} \)
19 \( 1 + (-2.16 - 2.16i)T + 19iT^{2} \)
23 \( 1 + 5.08T + 23T^{2} \)
29 \( 1 + (1.25 + 1.25i)T + 29iT^{2} \)
31 \( 1 + 1.27T + 31T^{2} \)
37 \( 1 + (2.25 + 2.25i)T + 37iT^{2} \)
41 \( 1 + 8.52iT - 41T^{2} \)
43 \( 1 + (1.61 + 1.61i)T + 43iT^{2} \)
47 \( 1 - 2.53iT - 47T^{2} \)
53 \( 1 + (5.67 + 5.67i)T + 53iT^{2} \)
59 \( 1 + (-7.81 + 7.81i)T - 59iT^{2} \)
61 \( 1 + (3.46 + 3.46i)T + 61iT^{2} \)
67 \( 1 + (6.29 - 6.29i)T - 67iT^{2} \)
71 \( 1 + 11.3iT - 71T^{2} \)
73 \( 1 + 16.1T + 73T^{2} \)
79 \( 1 - 1.13T + 79T^{2} \)
83 \( 1 + (-3.75 + 3.75i)T - 83iT^{2} \)
89 \( 1 - 3.98iT - 89T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−10.46526366336169279171515615655, −9.497376525496071674740355155476, −8.629989691269742627198314817903, −7.64138300270897214478899906552, −7.33111509042411242862220173631, −6.24306017004730467557825297956, −5.03434887745498572616798476126, −3.59468213976838148607819568263, −2.28493723243015529251579216581, −1.74792720180921090165275889869, 1.66868118391424064635558454000, 2.89636579361407735612328311049, 4.28286844856937013715119255785, 4.85328543745431527205100485220, 5.76217442934185280130716011960, 7.54602507750555239527152340293, 8.291783778098810681280434677400, 8.835470676216682742485984894420, 9.790693776764342657552409011308, 10.28178449637382431306244553712

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