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Properties

Label 2-2880-40.19-c0-0-2
Degree $2$
Conductor $2880$
Sign $0.707 + 0.707i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
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  − i·5-s + 2·11-s − 25-s − 2i·29-s + 2i·31-s − 49-s − 2i·55-s + 2·59-s − 2i·79-s − 2i·101-s + ⋯
L(s)  = 1  − i·5-s + 2·11-s − 25-s − 2i·29-s + 2i·31-s − 49-s − 2i·55-s + 2·59-s − 2i·79-s − 2i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.372625100\)
\(L(\frac12)\) \(\approx\) \(1.372625100\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.783657369057900701714203101957, −8.369632655632688164969595061574, −7.33293890324554048660987345593, −6.51694156187280777557359985029, −5.87308593690787922081053337325, −4.85546578795245221006698399064, −4.18566350898398097738529693399, −3.42260252171495640139610525099, −1.94945329819346414096283335174, −1.03374240986226013542076540995, 1.37821401674214160593088021186, 2.48888372295722238178186200326, 3.60561356532925833715740743212, 4.04460761757371185325953530357, 5.27201042305672359288825950897, 6.25710475158211018712219614858, 6.71450425727859806420209845443, 7.37528481067036743958510333924, 8.315137878168003150003949697264, 9.172416982970883378687871897455

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