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Properties

Label 2-2380-595.242-c0-0-0
Degree $2$
Conductor $2380$
Sign $-0.405 + 0.914i$
Analytic cond. $1.18777$
Root an. cond. $1.08985$
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 + (−0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (1.36 − 0.366i)11-s + 0.999i·15-s + (0.5 − 0.866i)17-s − 0.999·21-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + i·27-s + (1 + i)29-s + (−1.36 − 0.366i)33-s + (−0.866 − 0.499i)35-s + (0.5 + 0.866i)37-s + (0.366 − 1.36i)47-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (−0.5 − 0.866i)5-s + (0.866 − 0.5i)7-s + (1.36 − 0.366i)11-s + 0.999i·15-s + (0.5 − 0.866i)17-s − 0.999·21-s + (−0.5 − 0.866i)23-s + (−0.499 + 0.866i)25-s + i·27-s + (1 + i)29-s + (−1.36 − 0.366i)33-s + (−0.866 − 0.499i)35-s + (0.5 + 0.866i)37-s + (0.366 − 1.36i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.405 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2380\)    =    \(2^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $-0.405 + 0.914i$
Analytic conductor: \(1.18777\)
Root analytic conductor: \(1.08985\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2380} (837, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2380,\ (\ :0),\ -0.405 + 0.914i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9130024997\)
\(L(\frac12)\) \(\approx\) \(0.9130024997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
13 \( 1 - iT^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1 - i)T + iT^{2} \)
31 \( 1 + (-0.866 + 0.5i)T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (1.36 - 0.366i)T + (0.866 - 0.5i)T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 - 0.5i)T^{2} \)
67 \( 1 + (0.866 - 0.5i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (1 + i)T + iT^{2} \)
89 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.738695762234061330593963015833, −8.229790070686177518896529886678, −7.24706687873156774693055629626, −6.65901378430686667340386318130, −5.77318160191011952772285708569, −4.92599008614071804441628388348, −4.30102591005454452173239061413, −3.28478029140572576540239651431, −1.51573507383796495794383090234, −0.796766442715945526127340485753, 1.56205874740269964577555394445, 2.78094287693098407116864880043, 4.09217156655785281732411023889, 4.40940605015943033306542149757, 5.70086677318436190526264742079, 6.06053099242675684511107980341, 7.04745471430968167662089604122, 7.894876571192519725163616500105, 8.505838067225583047027353518920, 9.628954699927713269625778819949

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