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Properties

Label 2-3120-3120.1403-c0-0-0
Degree $2$
Conductor $3120$
Sign $0.160 - 0.987i$
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 − i·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 + 1.00i·10-s + (−1.41 + 1.41i)11-s − 1.00·12-s − 13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (0.707 + 0.707i)20-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s − i·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 + 1.00i·10-s + (−1.41 + 1.41i)11-s − 1.00·12-s − 13-s + (0.707 + 0.707i)15-s − 1.00·16-s + (−0.707 + 0.707i)18-s + (0.707 + 0.707i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.160 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.160 - 0.987i$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ 0.160 - 0.987i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08001311124\)
\(L(\frac12)\) \(\approx\) \(0.08001311124\)
\(L(1)\) 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 \( 1 + iT \)
5 \( 1 + (0.707 - 0.707i)T \)
13 \( 1 + T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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 \( 1 + iT^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
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 \( 1 + 1.41iT - T^{2} \)
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

−9.132239149331490120062351175364, −7.954575735829854232384846786333, −7.46554800379681735534188505968, −6.86022692852626790301970256145, −6.02247436752145706597905117965, −5.02497559814467665523069253253, −4.47689487045865461026577282405, −3.15133503387050127331573933314, −2.59755952013263953890126295506, −1.79181793040799001112143620030, 0.03470692101782975450575558167, 2.66877223551908649558626449409, 3.29711769106704733293278413975, 4.21357507361282376990810345026, 4.89881529778348761917045359375, 5.46618164495475627925674743316, 6.09298793381735070424610294119, 7.47234283285814979286356156250, 7.84155657410644360851355554088, 8.711990554621557499994340081543

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