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Properties

Label 2-3240-360.157-c0-0-4
Degree $2$
Conductor $3240$
Sign $0.665 + 0.746i$
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 + (0.707 + 0.707i)5-s + (0.5 − 1.86i)7-s + (−0.707 − 0.707i)8-s + (−0.500 − 0.866i)10-s + (0.448 − 0.258i)11-s + (−0.965 + 1.67i)14-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)20-s + (−0.5 + 0.133i)22-s + 1.00i·25-s + (1.36 − 1.36i)28-s + (0.707 + 1.22i)29-s + (0.866 − 1.5i)31-s + (−0.258 − 0.965i)32-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 + 0.707i)5-s + (0.5 − 1.86i)7-s + (−0.707 − 0.707i)8-s + (−0.500 − 0.866i)10-s + (0.448 − 0.258i)11-s + (−0.965 + 1.67i)14-s + (0.500 + 0.866i)16-s + (0.258 + 0.965i)20-s + (−0.5 + 0.133i)22-s + 1.00i·25-s + (1.36 − 1.36i)28-s + (0.707 + 1.22i)29-s + (0.866 − 1.5i)31-s + (−0.258 − 0.965i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.665 + 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3240\)    =    \(2^{3} \cdot 3^{4} \cdot 5\)
Sign: $0.665 + 0.746i$
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: \((2,\ 3240,\ (\ :0),\ 0.665 + 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.028982289\)
\(L(\frac12)\) \(\approx\) \(1.028982289\)
\(L(1)\) 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 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \)
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 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.133 - 0.5i)T + (-0.866 - 0.5i)T^{2} \)
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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

−8.785734867521575575924996470041, −7.88851593250382652118963590244, −7.36486562642793852212055661437, −6.66992105486098243642740176573, −6.12799127822887835781792604599, −4.77815242308919273159875633000, −3.80067884290033006646568711474, −3.05628274731013734569808498525, −1.86505246661233513974736409779, −0.963825869263204437145253518519, 1.34691162980031969295494615062, 2.15177700900057327760142852010, 2.91360328539664797925187474296, 4.69393274413429851827668937164, 5.26181974392976317165341627953, 6.13363047882953057882832926060, 6.46107182450912397658889562223, 7.76176448587065021930418429080, 8.428976051206879703897091139439, 8.862755802331189204814945965751

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