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Properties

Label 2-3120-3120.2027-c0-0-5
Degree $2$
Conductor $3120$
Sign $0.987 - 0.160i$
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  + 2-s − 3-s + 4-s + i·5-s − 6-s + 8-s + 9-s + i·10-s + (1 − i)11-s − 12-s − i·13-s − i·15-s + 16-s + 18-s + i·20-s + ⋯
L(s)  = 1  + 2-s − 3-s + 4-s + i·5-s − 6-s + 8-s + 9-s + i·10-s + (1 − i)11-s − 12-s − i·13-s − i·15-s + 16-s + 18-s + i·20-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3120\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 13\)
Sign: $0.987 - 0.160i$
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: \((2,\ 3120,\ (\ :0),\ 0.987 - 0.160i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.942020861\)
\(L(\frac12)\) \(\approx\) \(1.942020861\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
5 \( 1 - iT \)
13 \( 1 + iT \)
good7 \( 1 - iT^{2} \)
11 \( 1 + (-1 + i)T - iT^{2} \)
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 \( 1 - 2iT - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + (-1 + i)T - iT^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-1 - i)T + iT^{2} \)
61 \( 1 + (1 - i)T - iT^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2T + T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - 2T + T^{2} \)
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

−8.874727636736137797639156850024, −7.72288634045536267311636444078, −7.15544376123027019367274466455, −6.31600476447832181583345611750, −5.96042953052918766299839862796, −5.22703701737261334150039922251, −4.14683783861498269452519621091, −3.50485489618095978913996616713, −2.59351227088200858406988510676, −1.21723083943508021765365270515, 1.32696882187564700418419324080, 2.06006925360479751380487572977, 3.76335263207762939493006363177, 4.39730039811628908245755804644, 4.85999652486853502074512177184, 5.73329796274109485794391837452, 6.44048880554466320918856203709, 7.06955587634096588770728201470, 7.80316161093026387727967412457, 9.028101814651339983260523664253

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