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Properties

Label 2-3120-3120.1403-c0-0-7
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9170406320\)
\(L(\frac12)\) \(\approx\) \(0.9170406320\)
\(L(1)\) 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

−8.736002348417794462604920975617, −8.013680263824924233869108651674, −7.06037571839040260586221830579, −6.57372480625346363071276386365, −5.72245124528145281391761941462, −5.37228229604546389526875704777, −4.09074671647151457808933431874, −2.62083761514459993846421886758, −1.60738948439222232303369630040, −0.73174698881786034115263731854, 1.67382059304371206078745496482, 2.53318429246720296807905275780, 3.39969180934933823074972378983, 4.31126464811834611174770933723, 4.93948764310188089129296910328, 6.21687650180838788583029712927, 6.92365146259754760478630432702, 7.64442273635747507876264160415, 8.688756908408976807728339825366, 9.483860471476790352456374352438

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