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Properties

Label 2-3360-840.389-c0-0-3
Degree $2$
Conductor $3360$
Sign $-0.553 + 0.832i$
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  + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)5-s + (0.5 − 0.866i)7-s + (0.499 + 0.866i)9-s + (0.866 − 1.5i)11-s + (0.499 + 0.866i)15-s + (−0.866 + 0.499i)21-s + (0.499 + 0.866i)25-s − 0.999i·27-s + 1.73·29-s + (0.5 − 0.866i)31-s + (−1.5 + 0.866i)33-s + (−0.866 + 0.499i)35-s − 0.999i·45-s + (−0.499 − 0.866i)49-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.866 − 0.5i)5-s + (0.5 − 0.866i)7-s + (0.499 + 0.866i)9-s + (0.866 − 1.5i)11-s + (0.499 + 0.866i)15-s + (−0.866 + 0.499i)21-s + (0.499 + 0.866i)25-s − 0.999i·27-s + 1.73·29-s + (0.5 − 0.866i)31-s + (−1.5 + 0.866i)33-s + (−0.866 + 0.499i)35-s − 0.999i·45-s + (−0.499 − 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3360\)    =    \(2^{5} \cdot 3 \cdot 5 \cdot 7\)
Sign: $-0.553 + 0.832i$
Analytic conductor: \(1.67685\)
Root analytic conductor: \(1.29493\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3360} (1649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3360,\ (\ :0),\ -0.553 + 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8366620528\)
\(L(\frac12)\) \(\approx\) \(0.8366620528\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
good11 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - 1.73T + T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−8.264501010638080199733171603693, −7.908922027435803710220007461743, −6.99506878952822744349492779191, −6.37221143169905210160289259702, −5.55809908010840755912746469641, −4.59963254923658048700643713882, −4.11208642425419700829332585898, −3.05704747906475848402618881687, −1.39409594292326348562485309073, −0.66453768151723450173260740089, 1.41460542831912759723041278259, 2.70824659664764256926862783658, 3.76457067708149087298128149499, 4.66477425202364089352975642226, 4.93504306228401354458381929783, 6.23417187957323142510369225024, 6.66833733965607632879500182989, 7.46528916300769229771959285437, 8.328832941154642119325179270451, 9.114480365908580940304765587796

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