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Properties

Label 2-3660-3660.1319-c0-0-7
Degree $2$
Conductor $3660$
Sign $-0.942 - 0.335i$
Analytic cond. $1.82657$
Root an. cond. $1.35150$
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  + (−0.987 + 0.156i)2-s + (0.453 − 0.891i)3-s + (0.951 − 0.309i)4-s + (−0.309 − 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.610 − 0.0966i)7-s + (−0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (0.453 + 0.891i)10-s + (0.156 − 0.987i)12-s + 0.618·14-s + (−0.987 − 0.156i)15-s + (0.809 − 0.587i)16-s + (0.707 + 0.707i)18-s + (−0.587 − 0.809i)20-s + (−0.363 + 0.5i)21-s + ⋯
L(s)  = 1  + (−0.987 + 0.156i)2-s + (0.453 − 0.891i)3-s + (0.951 − 0.309i)4-s + (−0.309 − 0.951i)5-s + (−0.309 + 0.951i)6-s + (−0.610 − 0.0966i)7-s + (−0.891 + 0.453i)8-s + (−0.587 − 0.809i)9-s + (0.453 + 0.891i)10-s + (0.156 − 0.987i)12-s + 0.618·14-s + (−0.987 − 0.156i)15-s + (0.809 − 0.587i)16-s + (0.707 + 0.707i)18-s + (−0.587 − 0.809i)20-s + (−0.363 + 0.5i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.942 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3660\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 61\)
Sign: $-0.942 - 0.335i$
Analytic conductor: \(1.82657\)
Root analytic conductor: \(1.35150\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3660} (1319, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3660,\ (\ :0),\ -0.942 - 0.335i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3957554381\)
\(L(\frac12)\) \(\approx\) \(0.3957554381\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.987 - 0.156i)T \)
3 \( 1 + (-0.453 + 0.891i)T \)
5 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
good7 \( 1 + (0.610 + 0.0966i)T + (0.951 + 0.309i)T^{2} \)
11 \( 1 - iT^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.587 + 0.809i)T^{2} \)
19 \( 1 + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (1.04 + 0.533i)T + (0.587 + 0.809i)T^{2} \)
29 \( 1 + (0.642 + 0.642i)T + iT^{2} \)
31 \( 1 + (0.951 + 0.309i)T^{2} \)
37 \( 1 + (0.951 + 0.309i)T^{2} \)
41 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 + (0.280 + 0.550i)T + (-0.587 + 0.809i)T^{2} \)
47 \( 1 - 1.97iT - T^{2} \)
53 \( 1 + (0.587 - 0.809i)T^{2} \)
59 \( 1 + (0.951 - 0.309i)T^{2} \)
67 \( 1 + (1.78 + 0.907i)T + (0.587 + 0.809i)T^{2} \)
71 \( 1 + (-0.587 + 0.809i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-0.587 + 0.809i)T^{2} \)
83 \( 1 + (-0.183 + 0.253i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.221 + 1.39i)T + (-0.951 + 0.309i)T^{2} \)
97 \( 1 + (0.309 - 0.951i)T^{2} \)
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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

−8.218700613097898580989754279776, −7.72869466179484518854174448643, −7.12507660537225249346326226828, −6.12922227585982839015625028087, −5.78633021066435555910364761520, −4.39799899939549483689050595735, −3.36887064364635224641063884036, −2.36531452400742846226257682138, −1.44793102268574401361419396068, −0.28386011189850469716085362438, 1.92404079588518336532756051468, 2.87075200997036350202642165432, 3.43290773008435984450825117772, 4.21591007262767280046618612682, 5.57968497395459986454307920363, 6.29494368638589992513003391921, 7.12647145617619750029241019638, 7.82665565623283133234073490075, 8.408431830183282709012983165240, 9.329723583084303095702227443146

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