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Properties

Label 2-2880-48.35-c1-0-5
Degree $2$
Conductor $2880$
Sign $-0.220 - 0.975i$
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 − 4.87·7-s + (3.44 − 3.44i)11-s + (−3.87 − 3.87i)13-s + 1.41i·17-s + (1.87 − 1.87i)19-s + 5.47i·23-s + 1.00i·25-s + (−2.82 + 2.82i)29-s + 4i·31-s + (−3.44 − 3.44i)35-s + (−1 + i)37-s − 0.179·41-s + (6 + 6i)43-s − 1.41·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s − 1.84·7-s + (1.03 − 1.03i)11-s + (−1.07 − 1.07i)13-s + 0.342i·17-s + (0.429 − 0.429i)19-s + 1.14i·23-s + 0.200i·25-s + (−0.525 + 0.525i)29-s + 0.718i·31-s + (−0.582 − 0.582i)35-s + (−0.164 + 0.164i)37-s − 0.0280·41-s + (0.914 + 0.914i)43-s − 0.206·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.220 - 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7879456288\)
\(L(\frac12)\) \(\approx\) \(0.7879456288\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.87T + 7T^{2} \)
11 \( 1 + (-3.44 + 3.44i)T - 11iT^{2} \)
13 \( 1 + (3.87 + 3.87i)T + 13iT^{2} \)
17 \( 1 - 1.41iT - 17T^{2} \)
19 \( 1 + (-1.87 + 1.87i)T - 19iT^{2} \)
23 \( 1 - 5.47iT - 23T^{2} \)
29 \( 1 + (2.82 - 2.82i)T - 29iT^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 0.179T + 41T^{2} \)
43 \( 1 + (-6 - 6i)T + 43iT^{2} \)
47 \( 1 + 1.41T + 47T^{2} \)
53 \( 1 + (-1.41 - 1.41i)T + 53iT^{2} \)
59 \( 1 + (-3.62 + 3.62i)T - 59iT^{2} \)
61 \( 1 + (-4.74 - 4.74i)T + 61iT^{2} \)
67 \( 1 + (10.8 - 10.8i)T - 67iT^{2} \)
71 \( 1 + 1.23iT - 71T^{2} \)
73 \( 1 - 9.74iT - 73T^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + 83iT^{2} \)
89 \( 1 + 16.7T + 89T^{2} \)
97 \( 1 + 13.4T + 97T^{2} \)
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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

−9.210318215633436535057956082793, −8.321165844704135990113986980651, −7.17889242734271493984804496899, −6.83052858089342459779703722317, −5.83665962014727291341931172093, −5.50338160441441341910276201722, −4.01542646580699559475983352232, −3.20996815218362384284670105639, −2.76170817209230098944136982393, −1.08288695463009913161470086509, 0.27487396660688274398089548481, 1.88738765909803978358125204383, 2.74472319154937010515450495319, 3.90623489483020389395054882516, 4.46363565804606497462303713928, 5.58916811102565048296775859586, 6.46213901801699648196612727090, 6.88600372615489068297210662302, 7.58828283322604020301192337541, 8.932119121875606719332333498176

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