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Properties

Label 2-3240-360.13-c0-0-2
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 + (0.258 + 0.965i)5-s + (1.36 + 0.366i)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 + (0.258 − 0.965i)20-s + (0.366 + 1.36i)22-s + (−0.866 + 0.499i)25-s + (−1 − 0.999i)28-s + (0.707 + 1.22i)29-s + (0.965 − 0.258i)32-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.258 + 0.965i)5-s + (1.36 + 0.366i)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 + (0.258 − 0.965i)20-s + (0.366 + 1.36i)22-s + (−0.866 + 0.499i)25-s + (−1 − 0.999i)28-s + (0.707 + 1.22i)29-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

\(L(\frac{1}{2})\) \(\approx\) \(1.370046384\)
\(L(\frac12)\) \(\approx\) \(1.370046384\)
\(L(1)\) 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 \( 1 + (-0.258 - 0.965i)T \)
good7 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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 \( 1 + (-1.93 - 0.517i)T + (0.866 + 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
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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

−8.955980297875739591332255231423, −8.134587711347377194370612488674, −7.53828570966796092115190576017, −6.53133679058063381367358482886, −5.43975689545087039892601191993, −5.05136575208463753262704008308, −4.16375955167265504217714028596, −3.01711434556599858863824667237, −2.35846021153604624033132705705, −1.55757538134446899367419591286, 0.792687160217920063518940958737, 2.21768763276028734119083514099, 3.55414348873629007846544864172, 4.63609261035448884082700400868, 4.92842179595694169303867899667, 5.69548111480082388638887501235, 6.42369272670643193351501807933, 7.71373887910009107057634718136, 7.916294508539613027433695426847, 8.519060525993900127520000455042

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