Properties

Label 2-2880-16.5-c1-0-4
Degree $2$
Conductor $2880$
Sign $-0.468 - 0.883i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
Motivic weight $1$
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)5-s − 1.69i·7-s + (−1.72 − 1.72i)11-s + (0.217 − 0.217i)13-s − 3.27·17-s + (−1.73 + 1.73i)19-s + 2.93i·23-s + 1.00i·25-s + (−6.42 + 6.42i)29-s − 7.75·31-s + (1.20 − 1.20i)35-s + (4.70 + 4.70i)37-s + 4.36i·41-s + (4.30 + 4.30i)43-s + 0.568·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s − 0.642i·7-s + (−0.521 − 0.521i)11-s + (0.0602 − 0.0602i)13-s − 0.795·17-s + (−0.397 + 0.397i)19-s + 0.611i·23-s + 0.200i·25-s + (−1.19 + 1.19i)29-s − 1.39·31-s + (0.203 − 0.203i)35-s + (0.772 + 0.772i)37-s + 0.681i·41-s + (0.656 + 0.656i)43-s + 0.0829·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.468 - 0.883i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.468 - 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8325967997\)
\(L(\frac12)\) \(\approx\) \(0.8325967997\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + 1.69iT - 7T^{2} \)
11 \( 1 + (1.72 + 1.72i)T + 11iT^{2} \)
13 \( 1 + (-0.217 + 0.217i)T - 13iT^{2} \)
17 \( 1 + 3.27T + 17T^{2} \)
19 \( 1 + (1.73 - 1.73i)T - 19iT^{2} \)
23 \( 1 - 2.93iT - 23T^{2} \)
29 \( 1 + (6.42 - 6.42i)T - 29iT^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 + (-4.70 - 4.70i)T + 37iT^{2} \)
41 \( 1 - 4.36iT - 41T^{2} \)
43 \( 1 + (-4.30 - 4.30i)T + 43iT^{2} \)
47 \( 1 - 0.568T + 47T^{2} \)
53 \( 1 + (-0.749 - 0.749i)T + 53iT^{2} \)
59 \( 1 + (-2.21 - 2.21i)T + 59iT^{2} \)
61 \( 1 + (-7.39 + 7.39i)T - 61iT^{2} \)
67 \( 1 + (9.00 - 9.00i)T - 67iT^{2} \)
71 \( 1 - 2.32iT - 71T^{2} \)
73 \( 1 + 0.285iT - 73T^{2} \)
79 \( 1 - 2.59T + 79T^{2} \)
83 \( 1 + (10.7 - 10.7i)T - 83iT^{2} \)
89 \( 1 - 10.1iT - 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−9.059116700500186496596052643167, −8.231347076918941182327180774728, −7.44586347826413106790648176300, −6.83942075212165831094827991780, −5.92309043698836551473121722533, −5.28472687387283908854575417278, −4.21028784497434313293237211136, −3.44852988774446617952506443748, −2.45015196516421639823574856235, −1.32419331628571450898823855802, 0.25454877451884064930347777072, 2.00270599861529440902507045028, 2.46668326244713309435348127712, 3.86629874040993334380173068077, 4.62280062036637155963427812343, 5.54421822677212787445419988673, 6.06036732249089952993059461667, 7.11238133892619608961924167333, 7.70824550873879230626463246336, 8.831526349675441120462800492069

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