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Properties

Label 2-2880-16.5-c1-0-36
Degree $2$
Conductor $2880$
Sign $-0.596 + 0.802i$
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 − 4.50i·7-s + (−1.64 − 1.64i)11-s + (1.51 − 1.51i)13-s − 1.45·17-s + (2.67 − 2.67i)19-s + 2.37i·23-s + 1.00i·25-s + (−0.924 + 0.924i)29-s + 7.20·31-s + (3.18 − 3.18i)35-s + (−5.21 − 5.21i)37-s + 6.41i·41-s + (−7.65 − 7.65i)43-s − 2.51·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s − 1.70i·7-s + (−0.494 − 0.494i)11-s + (0.421 − 0.421i)13-s − 0.353·17-s + (0.614 − 0.614i)19-s + 0.495i·23-s + 0.200i·25-s + (−0.171 + 0.171i)29-s + 1.29·31-s + (0.539 − 0.539i)35-s + (−0.856 − 0.856i)37-s + 1.00i·41-s + (−1.16 − 1.16i)43-s − 0.366·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.596 + 0.802i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.596 + 0.802i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.362165476\)
\(L(\frac12)\) \(\approx\) \(1.362165476\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.50iT - 7T^{2} \)
11 \( 1 + (1.64 + 1.64i)T + 11iT^{2} \)
13 \( 1 + (-1.51 + 1.51i)T - 13iT^{2} \)
17 \( 1 + 1.45T + 17T^{2} \)
19 \( 1 + (-2.67 + 2.67i)T - 19iT^{2} \)
23 \( 1 - 2.37iT - 23T^{2} \)
29 \( 1 + (0.924 - 0.924i)T - 29iT^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 + (5.21 + 5.21i)T + 37iT^{2} \)
41 \( 1 - 6.41iT - 41T^{2} \)
43 \( 1 + (7.65 + 7.65i)T + 43iT^{2} \)
47 \( 1 + 2.51T + 47T^{2} \)
53 \( 1 + (1.50 + 1.50i)T + 53iT^{2} \)
59 \( 1 + (5.31 + 5.31i)T + 59iT^{2} \)
61 \( 1 + (1.02 - 1.02i)T - 61iT^{2} \)
67 \( 1 + (5.22 - 5.22i)T - 67iT^{2} \)
71 \( 1 + 1.92iT - 71T^{2} \)
73 \( 1 + 1.39iT - 73T^{2} \)
79 \( 1 + 5.06T + 79T^{2} \)
83 \( 1 + (2.44 - 2.44i)T - 83iT^{2} \)
89 \( 1 + 9.36iT - 89T^{2} \)
97 \( 1 - 18.6T + 97T^{2} \)
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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

−8.389164386421136865334022185975, −7.68060840151157327195558805289, −7.02261462081060597572055579796, −6.37077120638181028491873460125, −5.38328346252469310546286885140, −4.57837186445458079694176746110, −3.60636348238246419381960454113, −2.96557378684797944271998065638, −1.53969687115054165341599489241, −0.42618359424840139170012611028, 1.54680650143038956932655398245, 2.39663912748332025163492894774, 3.24940575716627246974595489124, 4.58603052812728292685387320988, 5.16839824024630085549478033449, 6.01024084630045047389254706816, 6.53151906298820664442041033754, 7.68295586122210606387849981693, 8.480349854787992615843093908860, 8.903468579237656101638660760196

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