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Properties

Label 2-2880-16.5-c1-0-6
Degree $2$
Conductor $2880$
Sign $-0.857 - 0.513i$
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 + 1.41i·7-s + (1.11 + 1.11i)11-s + (−0.271 + 0.271i)13-s − 0.744·17-s + (−5.21 + 5.21i)19-s + 4.76i·23-s + 1.00i·25-s + (1.21 − 1.21i)29-s − 7.75·31-s + (−1.00 + 1.00i)35-s + (−5.32 − 5.32i)37-s − 7.33i·41-s + (−6.78 − 6.78i)43-s − 0.735·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 0.534i·7-s + (0.335 + 0.335i)11-s + (−0.0752 + 0.0752i)13-s − 0.180·17-s + (−1.19 + 1.19i)19-s + 0.994i·23-s + 0.200i·25-s + (0.225 − 0.225i)29-s − 1.39·31-s + (−0.169 + 0.169i)35-s + (−0.875 − 0.875i)37-s − 1.14i·41-s + (−1.03 − 1.03i)43-s − 0.107·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.857 - 0.513i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.857 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9705404727\)
\(L(\frac12)\) \(\approx\) \(0.9705404727\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.41iT - 7T^{2} \)
11 \( 1 + (-1.11 - 1.11i)T + 11iT^{2} \)
13 \( 1 + (0.271 - 0.271i)T - 13iT^{2} \)
17 \( 1 + 0.744T + 17T^{2} \)
19 \( 1 + (5.21 - 5.21i)T - 19iT^{2} \)
23 \( 1 - 4.76iT - 23T^{2} \)
29 \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 + (5.32 + 5.32i)T + 37iT^{2} \)
41 \( 1 + 7.33iT - 41T^{2} \)
43 \( 1 + (6.78 + 6.78i)T + 43iT^{2} \)
47 \( 1 + 0.735T + 47T^{2} \)
53 \( 1 + (-9.55 - 9.55i)T + 53iT^{2} \)
59 \( 1 + (-1.62 - 1.62i)T + 59iT^{2} \)
61 \( 1 + (5.70 - 5.70i)T - 61iT^{2} \)
67 \( 1 + (5.59 - 5.59i)T - 67iT^{2} \)
71 \( 1 - 8.60iT - 71T^{2} \)
73 \( 1 - 4.28iT - 73T^{2} \)
79 \( 1 - 1.01T + 79T^{2} \)
83 \( 1 + (-1.68 + 1.68i)T - 83iT^{2} \)
89 \( 1 + 10.3iT - 89T^{2} \)
97 \( 1 + 16.9T + 97T^{2} \)
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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

−8.934123502298446741993890188847, −8.567682166076079885532620067008, −7.38413371437254390941265112129, −6.95605613480376810350030054250, −5.78533620778816433701934489028, −5.57610119801193142933817433300, −4.23985945144549670551248074842, −3.58876961664368826118996462457, −2.34845442023142165385037855384, −1.65627883276605767004618296716, 0.28907143577517498126561050314, 1.58246635530789511288800931317, 2.66565667665013159096075050049, 3.71359489089577248743375242698, 4.61264393575499268111078795513, 5.21677956796211185067411108062, 6.48269503530446119920131940103, 6.66362597684304131986076684984, 7.78163891698641107768759110778, 8.580605243642652497609362024011

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