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Properties

Label 2-3120-3120.1949-c0-0-4
Degree $2$
Conductor $3120$
Sign $1$
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  + (−0.382 + 0.923i)2-s + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (−0.382 + 0.923i)5-s + (0.382 + 0.923i)6-s + (0.923 − 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (0.541 − 0.541i)11-s − 12-s + (0.707 − 0.707i)13-s + (0.382 + 0.923i)15-s + i·16-s + (0.923 + 0.382i)18-s + (0.923 − 0.382i)20-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s + (0.707 − 0.707i)3-s + (−0.707 − 0.707i)4-s + (−0.382 + 0.923i)5-s + (0.382 + 0.923i)6-s + (0.923 − 0.382i)8-s − 1.00i·9-s + (−0.707 − 0.707i)10-s + (0.541 − 0.541i)11-s − 12-s + (0.707 − 0.707i)13-s + (0.382 + 0.923i)15-s + i·16-s + (0.923 + 0.382i)18-s + (0.923 − 0.382i)20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $1$
Analytic conductor: \(1.55708\)
Root analytic conductor: \(1.24783\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3120} (1949, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3120,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.181422592\)
\(L(\frac12)\) \(\approx\) \(1.181422592\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.382 - 0.923i)T \)
13 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + (-0.541 + 0.541i)T - iT^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - iT^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 0.765iT - T^{2} \)
43 \( 1 + (1 + i)T + iT^{2} \)
47 \( 1 - 0.765iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-1.30 + 1.30i)T - iT^{2} \)
61 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.84iT - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (0.541 - 0.541i)T - iT^{2} \)
89 \( 1 - 1.84iT - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.572464515004053945873891731340, −8.117314894801526911320593930511, −7.37663611162788620491638047733, −6.74501880158963238035819814299, −6.19188500407809544389729176135, −5.38157083459993828137543327013, −3.93824279104559565000569606392, −3.45083293884318961149767578561, −2.22557599107468987828731577398, −0.880245963443038518645194146403, 1.31771773893373778788949237232, 2.22340684437580297933532424171, 3.41365982391480529478809533148, 4.07365409139314287531633374146, 4.60147585679988702295418452677, 5.45588974383934522100449806625, 6.87868404275405238977825878294, 7.76991668203894426356591088953, 8.578104943485518306014697773977, 8.785128377453124425297204029805

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