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Properties

Label 2-3640-3640.1077-c0-0-3
Degree $2$
Conductor $3640$
Sign $0.439 - 0.898i$
Analytic cond. $1.81659$
Root an. cond. $1.34781$
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 + (1.86 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (−1.86 + 0.5i)6-s + (−0.5 + 0.866i)7-s + 0.999i·8-s + (2.36 + 1.36i)9-s + (−0.499 + 0.866i)10-s + (1.36 − 1.36i)12-s + (−0.5 + 0.866i)13-s − 0.999i·14-s + (1.86 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 2.73·18-s + (−0.366 − 1.36i)19-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (1.86 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.866 − 0.5i)5-s + (−1.86 + 0.5i)6-s + (−0.5 + 0.866i)7-s + 0.999i·8-s + (2.36 + 1.36i)9-s + (−0.499 + 0.866i)10-s + (1.36 − 1.36i)12-s + (−0.5 + 0.866i)13-s − 0.999i·14-s + (1.86 − 0.5i)15-s + (−0.5 − 0.866i)16-s − 2.73·18-s + (−0.366 − 1.36i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3640 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3640\)    =    \(2^{3} \cdot 5 \cdot 7 \cdot 13\)
Sign: $0.439 - 0.898i$
Analytic conductor: \(1.81659\)
Root analytic conductor: \(1.34781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3640} (1077, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3640,\ (\ :0),\ 0.439 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.864028496\)
\(L(\frac12)\) \(\approx\) \(1.864028496\)
\(L(1)\) 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 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-1.86 - 0.5i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.866 - 0.5i)T^{2} \)
17 \( 1 + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.366 + 1.36i)T + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - iT^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (-0.5 + 0.133i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (1.86 - 0.5i)T + (0.866 - 0.5i)T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.760428135744426500113706357592, −8.682427724779613157831766410451, −7.51322985855619983490966944667, −6.90205905357024084429111450491, −6.05459334021585686754165626970, −4.97868518664218684540426056658, −4.38998974699965501608107620147, −2.88134454419083045217568568389, −2.42201797936048517226690472070, −1.63057996638648079592946353468, 1.28893384158725142045425249947, 2.04724925094137898348553547663, 2.97325360683199416188129026161, 3.37852675183738250405878977166, 4.25207270475364403009862908641, 5.95478992918433193200905813107, 6.85946669044256565798368205595, 7.33828943964296382125448039230, 7.958685656200726417963807984802, 8.563195026069908718332824329033

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