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Properties

Label 2-3240-360.13-c0-0-1
Degree $2$
Conductor $3240$
Sign $0.989 - 0.144i$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.965 − 0.258i)5-s + (−1.86 − 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.499 + 0.866i)10-s + (−0.448 + 0.258i)11-s + (−0.965 + 1.67i)14-s + (0.500 + 0.866i)16-s + (0.707 + 0.707i)20-s + (0.133 + 0.5i)22-s + (0.866 + 0.499i)25-s + (1.36 + 1.36i)28-s + (0.707 + 1.22i)29-s + (0.866 − 1.5i)31-s + (0.965 − 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (−0.965 − 0.258i)5-s + (−1.86 − 0.5i)7-s + (−0.707 + 0.707i)8-s + (−0.499 + 0.866i)10-s + (−0.448 + 0.258i)11-s + (−0.965 + 1.67i)14-s + (0.500 + 0.866i)16-s + (0.707 + 0.707i)20-s + (0.133 + 0.5i)22-s + (0.866 + 0.499i)25-s + (1.36 + 1.36i)28-s + (0.707 + 1.22i)29-s + (0.866 − 1.5i)31-s + (0.965 − 0.258i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 - 0.144i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.989 - 0.144i$
Analytic conductor: \(1.61697\)
Root analytic conductor: \(1.27160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3240} (2053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.989 - 0.144i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4007688483\)
\(L(\frac12)\) \(\approx\) \(0.4007688483\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (0.965 + 0.258i)T \)
good7 \( 1 + (1.86 + 0.5i)T + (0.866 + 0.5i)T^{2} \)
11 \( 1 + (0.448 - 0.258i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (1.22 - 1.22i)T - iT^{2} \)
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 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.965 + 0.258i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 - 0.133i)T + (0.866 + 0.5i)T^{2} \)
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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

−9.123474263176358335351929640667, −8.222377301173236602873715310323, −7.38804258965619775420507250242, −6.55526684847745111817767022012, −5.76171322569431676326750823024, −4.66754189997827644000752836951, −4.04636740571956194626881078142, −3.23816959659577452394131258613, −2.67139501853315424500733875342, −0.976945980040680551087506228079, 0.28073863328178095800676046063, 2.86554423627442442801256441509, 3.26596121689445653782724381515, 4.17267534995221109970395120396, 5.07951089396949614697944045632, 6.03969882538878391483379490573, 6.59543705709748662907808709610, 7.11197263074427481791538967206, 8.134935965379482375786806624357, 8.507216531244278020176976420290

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