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Properties

Label 2-2880-15.2-c1-0-35
Degree $2$
Conductor $2880$
Sign $-0.333 + 0.942i$
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  + (−0.137 − 2.23i)5-s + (−0.806 − 0.806i)7-s + 5.87i·11-s + (0.193 − 0.193i)13-s + (3.50 − 3.50i)17-s − 2.38i·19-s + (−4.18 − 4.18i)23-s + (−4.96 + 0.612i)25-s + 7.29·29-s + 4.31·31-s + (−1.68 + 1.90i)35-s + (−1.54 − 1.54i)37-s − 3.32i·41-s + (−8.31 + 8.31i)43-s + (6.42 − 6.42i)47-s + ⋯
L(s)  = 1  + (−0.0613 − 0.998i)5-s + (−0.304 − 0.304i)7-s + 1.77i·11-s + (0.0537 − 0.0537i)13-s + (0.851 − 0.851i)17-s − 0.547i·19-s + (−0.873 − 0.873i)23-s + (−0.992 + 0.122i)25-s + 1.35·29-s + 0.774·31-s + (−0.285 + 0.322i)35-s + (−0.253 − 0.253i)37-s − 0.519i·41-s + (−1.26 + 1.26i)43-s + (0.937 − 0.937i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.333 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.333 + 0.942i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.333 + 0.942i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.312374069\)
\(L(\frac12)\) \(\approx\) \(1.312374069\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.137 + 2.23i)T \)
good7 \( 1 + (0.806 + 0.806i)T + 7iT^{2} \)
11 \( 1 - 5.87iT - 11T^{2} \)
13 \( 1 + (-0.193 + 0.193i)T - 13iT^{2} \)
17 \( 1 + (-3.50 + 3.50i)T - 17iT^{2} \)
19 \( 1 + 2.38iT - 19T^{2} \)
23 \( 1 + (4.18 + 4.18i)T + 23iT^{2} \)
29 \( 1 - 7.29T + 29T^{2} \)
31 \( 1 - 4.31T + 31T^{2} \)
37 \( 1 + (1.54 + 1.54i)T + 37iT^{2} \)
41 \( 1 + 3.32iT - 41T^{2} \)
43 \( 1 + (8.31 - 8.31i)T - 43iT^{2} \)
47 \( 1 + (-6.42 + 6.42i)T - 47iT^{2} \)
53 \( 1 + (4.64 + 4.64i)T + 53iT^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 7.53T + 61T^{2} \)
67 \( 1 + (4.31 + 4.31i)T + 67iT^{2} \)
71 \( 1 + 9.47iT - 71T^{2} \)
73 \( 1 + (-0.0376 + 0.0376i)T - 73iT^{2} \)
79 \( 1 - 3.68iT - 79T^{2} \)
83 \( 1 + (-0.221 - 0.221i)T + 83iT^{2} \)
89 \( 1 - 9.52T + 89T^{2} \)
97 \( 1 + (8.27 + 8.27i)T + 97iT^{2} \)
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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

−8.499475035474925199684006307261, −7.84183678110083064767512051476, −7.04823456244173977911912182928, −6.37895482077572664178661488854, −5.17211625264416743626040911290, −4.72336269107035606538562326226, −3.96137586676295266727725328924, −2.72826814931873572134888980290, −1.67682981630158909270522919179, −0.44583210892901465036861115012, 1.24538404620366401573694352498, 2.65753967236408114909206162340, 3.34050767517342631522088638455, 3.99104302043243057717066754150, 5.41955283845125890431465331720, 6.10101859877677740319204229157, 6.46189793734531098643979243167, 7.64799658550961426712208829159, 8.180975129848166415772230737917, 8.875398517835552440789337730874

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