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Properties

Label 2-3240-360.133-c0-0-5
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: $\chi_{3240} (3133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3240,\ (\ :0),\ 0.313 + 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.492004059\)
\(L(\frac12)\) \(\approx\) \(2.492004059\)
\(L(1)\) 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

−8.715627932279226063440270125090, −7.59604481464195046311521501819, −7.06652461968233570463027023588, −6.31988985650434796274683056495, −5.53982358293108620969233557391, −4.95793071709419393104068867564, −3.87611075978357566350587448312, −3.16804302920859564713553089084, −2.32238080484816056111243537569, −1.10704796953701191815317559396, 2.06263961668479388415381034662, 2.38580947842421631857438662720, 3.37795632696014827589804148908, 4.67907034754980612577591469854, 5.22093771520546541652741934447, 5.88447303804825415756475821098, 6.38292137359405815140463244759, 7.40210366590670408441488654405, 8.135701605588020097364400032070, 8.951212336139925638881089175893

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