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Properties

Label 2-2880-40.19-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.6699649747\)
\(L(\frac12)\) \(\approx\) \(0.6699649747\)
\(L(1)\) 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

−8.475054487925109906245152847210, −7.949759081703459017712212768714, −7.50341078341659257465136480910, −6.13492548583102592963685286850, −5.59410818043879712713621234199, −4.77873223941865013788183837483, −4.11531973979706627298228323569, −2.82405056728476985418503660155, −1.99146617552321603127566000554, −0.38898812579356765423182752513, 1.78842535586474670052568472846, 2.94774358625620408474538972010, 3.29257989099249375954507285473, 4.76319198064855138069403366956, 5.31811541550600676038158744916, 6.26646123443303154053973883447, 7.07518669611979680886792163217, 7.64280772688851557876122890586, 8.375646719949536093022274072792, 9.228213744599735915277949965899

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