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Properties

Label 2-3360-840.389-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(2.015461278\)
\(L(\frac12)\) \(\approx\) \(2.015461278\)
\(L(1)\) 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

−9.044651534568144376897532817426, −8.076853912668849779704105368485, −7.39751700102035606303451319933, −7.04061963329356884778400726219, −5.78445422622444924915063157382, −4.94419933224220499746627567745, −4.30255975086990826350758572686, −3.37828360226792300866770150108, −2.30613873186312757674385410765, −1.77898960763403525164835240763, 1.16033213190011212783963454945, 2.21405427560329911039044911435, 2.83129505915355814936910632496, 3.83607211641063197092203292660, 5.11261790111566541857994766914, 5.64864481010685982334220026183, 6.32519363628123051053998326278, 7.35566419338377136988310591336, 8.208262147347088806641447228895, 8.626838608420905529632864379369

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