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Properties

Label 2-3360-840.677-c0-0-0
Degree $2$
Conductor $3360$
Sign $0.350 - 0.936i$
Analytic cond. $1.67685$
Root an. cond. $1.29493$
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)3-s + (−0.965 − 0.258i)5-s + (−0.5 − 0.866i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.448i)11-s + (0.499 − 0.866i)15-s + (0.965 − 0.258i)21-s + (0.866 + 0.499i)25-s + (0.707 − 0.707i)27-s + 1.93i·29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.133i)33-s + (0.258 + 0.965i)35-s + (0.707 + 0.707i)45-s + (−0.499 + 0.866i)49-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)3-s + (−0.965 − 0.258i)5-s + (−0.5 − 0.866i)7-s + (−0.866 − 0.499i)9-s + (0.258 + 0.448i)11-s + (0.499 − 0.866i)15-s + (0.965 − 0.258i)21-s + (0.866 + 0.499i)25-s + (0.707 − 0.707i)27-s + 1.93i·29-s + (0.866 − 0.5i)31-s + (−0.5 + 0.133i)33-s + (0.258 + 0.965i)35-s + (0.707 + 0.707i)45-s + (−0.499 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.350 - 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $0.350 - 0.936i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1937, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ 0.350 - 0.936i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7683601820\)
\(L(\frac12)\) \(\approx\) \(0.7683601820\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
good11 \( 1 + (-0.258 - 0.448i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - iT^{2} \)
17 \( 1 + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 - 1.93iT - T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
59 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.366 - 1.36i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
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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

−8.913828037219160168513222138533, −8.383724973073473972414978260758, −7.31240657508207687212439580998, −6.88213426195732256257377227841, −5.82492401420018382434727343847, −4.91369236859263187156681300941, −4.23039298923111379506419831337, −3.69199422589384041643170782031, −2.83341556542487973405343373702, −0.981462993389934631798421782235, 0.61370042393591396384106142902, 2.15821680228321594715019110246, 2.95272525346976724246834340601, 3.84034018215622305407963767222, 4.95094451867362589203650524354, 5.85748740158856928753823874906, 6.47303143640567397754072837915, 7.08162167721271398590571731456, 8.057607220881345432273619396166, 8.343116604974015590706709390643

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