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Properties

Label 2-3120-3120.2027-c0-0-4
Degree $2$
Conductor $3120$
Sign $0.584 - 0.811i$
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  + (−0.707 + 0.707i)2-s + 3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (0.707 + 0.707i)8-s + 9-s − 1.00·10-s − 1.00i·12-s − i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (0.707 − 0.707i)20-s + (0.707 + 0.707i)24-s + 1.00i·25-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + 3-s − 1.00i·4-s + (0.707 + 0.707i)5-s + (−0.707 + 0.707i)6-s + (0.707 + 0.707i)8-s + 9-s − 1.00·10-s − 1.00i·12-s − i·13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (0.707 − 0.707i)20-s + (0.707 + 0.707i)24-s + 1.00i·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.584 - 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.584 - 0.811i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (2027, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ 0.584 - 0.811i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.476998787\)
\(L(\frac12)\) \(\approx\) \(1.476998787\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.848762283689281379140287573330, −8.296265458465612065796222174080, −7.47258355996885764974075670291, −7.00708991612071726924951765775, −6.10804975842620779362656997038, −5.43258732665059010565351648712, −4.37853535239591614502332316656, −3.16474178869308125783964334162, −2.39755452717361779660028597755, −1.34795475522225406276137081947, 1.25600012921165273930164869197, 2.06937465498595576206894228063, 2.82478146593026054884867817788, 3.99590486567525974815403965615, 4.48811448504231746295313883959, 5.70997482577961223543582404032, 6.87558911581368782349431216103, 7.41963251869884032694322871353, 8.449531309508227574716681082487, 8.800821627957791258336647577133

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