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Properties

Label 2-3240-360.157-c0-0-1
Degree $2$
Conductor $3240$
Sign $0.313 - 0.949i$
Analytic cond. $1.61697$
Root an. cond. $1.27160$
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  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s + (−0.366 + 1.36i)7-s + (−0.707 − 0.707i)8-s + 10-s + (1.22 − 0.707i)11-s + (0.707 − 1.22i)14-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)20-s + (−1.36 + 0.366i)22-s + (0.866 − 0.499i)25-s + (−0.999 + i)28-s + (0.707 + 1.22i)29-s + (−0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (−0.965 + 0.258i)5-s + (−0.366 + 1.36i)7-s + (−0.707 − 0.707i)8-s + 10-s + (1.22 − 0.707i)11-s + (0.707 − 1.22i)14-s + (0.500 + 0.866i)16-s + (−0.965 − 0.258i)20-s + (−1.36 + 0.366i)22-s + (0.866 − 0.499i)25-s + (−0.999 + i)28-s + (0.707 + 1.22i)29-s + (−0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.313 - 0.949i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.313 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6114528197\)
\(L(\frac12)\) \(\approx\) \(0.6114528197\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
5 \( 1 + (0.965 - 0.258i)T \)
good7 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
11 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.866 - 0.5i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.707 - 1.22i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.517 - 1.93i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
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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

−8.965908768560566417363369078016, −8.434416190408242907969404098962, −7.70679348425701516257218004975, −6.70646485939256033800957915645, −6.35592335740712452198476124694, −5.31340557829523430115457298098, −3.99717076174471054249734159321, −3.24667732866982159247730238143, −2.52607811315494911474794114045, −1.18503271809914169823083332240, 0.59869785783696980048778198147, 1.64387900723532830879436677861, 3.11984715066829374975783568049, 4.05551225199205840567326758452, 4.64039077814051121448424462516, 5.97788034144162158318465905025, 6.90289011418694546729717078125, 7.11490587096408431199118296855, 7.980297574645901217200489856278, 8.544942771208657749708861014643

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