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Properties

Label 2-2880-16.13-c1-0-15
Degree $2$
Conductor $2880$
Sign $0.779 - 0.626i$
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 − 0.0588i·7-s + (2.23 − 2.23i)11-s + (2.84 + 2.84i)13-s − 5.98·17-s + (0.617 + 0.617i)19-s − 0.746i·23-s − 1.00i·25-s + (1.13 + 1.13i)29-s + 8.55·31-s + (0.0416 + 0.0416i)35-s + (−2.01 + 2.01i)37-s + 7.71i·41-s + (2.94 − 2.94i)43-s − 0.789·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s − 0.0222i·7-s + (0.673 − 0.673i)11-s + (0.790 + 0.790i)13-s − 1.45·17-s + (0.141 + 0.141i)19-s − 0.155i·23-s − 0.200i·25-s + (0.211 + 0.211i)29-s + 1.53·31-s + (0.00703 + 0.00703i)35-s + (−0.331 + 0.331i)37-s + 1.20i·41-s + (0.448 − 0.448i)43-s − 0.115·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.779 - 0.626i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.779 - 0.626i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.779 - 0.626i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.703853209\)
\(L(\frac12)\) \(\approx\) \(1.703853209\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.0588iT - 7T^{2} \)
11 \( 1 + (-2.23 + 2.23i)T - 11iT^{2} \)
13 \( 1 + (-2.84 - 2.84i)T + 13iT^{2} \)
17 \( 1 + 5.98T + 17T^{2} \)
19 \( 1 + (-0.617 - 0.617i)T + 19iT^{2} \)
23 \( 1 + 0.746iT - 23T^{2} \)
29 \( 1 + (-1.13 - 1.13i)T + 29iT^{2} \)
31 \( 1 - 8.55T + 31T^{2} \)
37 \( 1 + (2.01 - 2.01i)T - 37iT^{2} \)
41 \( 1 - 7.71iT - 41T^{2} \)
43 \( 1 + (-2.94 + 2.94i)T - 43iT^{2} \)
47 \( 1 + 0.789T + 47T^{2} \)
53 \( 1 + (-6.80 + 6.80i)T - 53iT^{2} \)
59 \( 1 + (9.36 - 9.36i)T - 59iT^{2} \)
61 \( 1 + (0.814 + 0.814i)T + 61iT^{2} \)
67 \( 1 + (5.46 + 5.46i)T + 67iT^{2} \)
71 \( 1 + 7.40iT - 71T^{2} \)
73 \( 1 - 11.6iT - 73T^{2} \)
79 \( 1 - 17.4T + 79T^{2} \)
83 \( 1 + (7.55 + 7.55i)T + 83iT^{2} \)
89 \( 1 - 16.3iT - 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.765140226655703631798615452772, −8.289864019931634354013902945152, −7.23936361943009777173614457775, −6.46442320653853549478615696242, −6.11848482189142446200862135190, −4.78711749109455627109438788136, −4.11463326378573849189848950746, −3.29263645674698116912715623209, −2.22703264813948511924712311218, −0.981311273651500022413172077233, 0.69038884459986485498418553257, 1.92316430084443792117743877222, 3.02965482060314648350598247415, 4.09836473573594870952479668819, 4.60750380584134155887781594277, 5.66832870274310758246295377648, 6.44177822478731507176173952794, 7.17751689453584915885157483546, 7.990662347294641962541439426575, 8.778579055725062978795194185731

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