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Properties

Label 2-3120-3120.1667-c0-0-1
Degree $2$
Conductor $3120$
Sign $0.909 - 0.416i$
Analytic cond. $1.55708$
Root an. cond. $1.24783$
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)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.258 − 0.965i)6-s + (1.36 − 0.366i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.965 + 0.258i)11-s + 12-s − 13-s + 1.41i·14-s + (−0.258 + 0.965i)15-s + (0.500 + 0.866i)16-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)5-s + (−0.258 − 0.965i)6-s + (1.36 − 0.366i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.500 + 0.866i)10-s + (0.965 + 0.258i)11-s + 12-s − 13-s + 1.41i·14-s + (−0.258 + 0.965i)15-s + (0.500 + 0.866i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.909 - 0.416i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1667, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ 0.909 - 0.416i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.017750908\)
\(L(\frac12)\) \(\approx\) \(1.017750908\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.707 + 0.707i)T \)
13 \( 1 + T \)
good7 \( 1 + (-1.36 + 0.366i)T + (0.866 - 0.5i)T^{2} \)
11 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 - 0.5i)T^{2} \)
23 \( 1 + (-0.258 + 0.965i)T + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.258 - 0.965i)T + (-0.866 - 0.5i)T^{2} \)
31 \( 1 + iT - T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + (-0.965 + 0.258i)T + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (0.866 - 0.5i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (1 + i)T + iT^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-1.22 + 0.707i)T + (0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T^{2} \)
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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

−9.105811523339530861513624025520, −8.157992024653544899942173736277, −7.31436690880026576386221590575, −6.65577052429856375802304064373, −5.78145280313983482639941401962, −5.12979130611906606961249875881, −4.59500990989726006774843083386, −4.01845438392333121378937289381, −1.90153826898612531152155393771, −0.893620945450583998486857209975, 1.35019809637097131371702078187, 1.93456459813039635676141710918, 2.90064673021697350532017591670, 4.17544784555076747853787742254, 5.05864090013959070908974549027, 5.56245378869241209403181477647, 6.60744125522638552511875287068, 7.40313511105713006476650913581, 8.059850448100512562881715679303, 8.973207206722663419329458286124

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