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Properties

Label 2-2880-16.13-c1-0-7
Degree $2$
Conductor $2880$
Sign $-0.488 - 0.872i$
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 + 4.06i·7-s + (−1.15 + 1.15i)11-s + (−2.48 − 2.48i)13-s + 3.58·17-s + (−4.19 − 4.19i)19-s + 4.42i·23-s − 1.00i·25-s + (2.76 + 2.76i)29-s + 10.7·31-s + (2.87 + 2.87i)35-s + (−4.11 + 4.11i)37-s + 10.9i·41-s + (−0.217 + 0.217i)43-s − 7.21·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + 1.53i·7-s + (−0.349 + 0.349i)11-s + (−0.689 − 0.689i)13-s + 0.869·17-s + (−0.963 − 0.963i)19-s + 0.923i·23-s − 0.200i·25-s + (0.513 + 0.513i)29-s + 1.92·31-s + (0.486 + 0.486i)35-s + (−0.675 + 0.675i)37-s + 1.71i·41-s + (−0.0331 + 0.0331i)43-s − 1.05·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 - 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.488 - 0.872i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.488 - 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.199989688\)
\(L(\frac12)\) \(\approx\) \(1.199989688\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.06iT - 7T^{2} \)
11 \( 1 + (1.15 - 1.15i)T - 11iT^{2} \)
13 \( 1 + (2.48 + 2.48i)T + 13iT^{2} \)
17 \( 1 - 3.58T + 17T^{2} \)
19 \( 1 + (4.19 + 4.19i)T + 19iT^{2} \)
23 \( 1 - 4.42iT - 23T^{2} \)
29 \( 1 + (-2.76 - 2.76i)T + 29iT^{2} \)
31 \( 1 - 10.7T + 31T^{2} \)
37 \( 1 + (4.11 - 4.11i)T - 37iT^{2} \)
41 \( 1 - 10.9iT - 41T^{2} \)
43 \( 1 + (0.217 - 0.217i)T - 43iT^{2} \)
47 \( 1 + 7.21T + 47T^{2} \)
53 \( 1 + (4.93 - 4.93i)T - 53iT^{2} \)
59 \( 1 + (9.20 - 9.20i)T - 59iT^{2} \)
61 \( 1 + (-1.03 - 1.03i)T + 61iT^{2} \)
67 \( 1 + (-0.201 - 0.201i)T + 67iT^{2} \)
71 \( 1 + 6.73iT - 71T^{2} \)
73 \( 1 - 1.83iT - 73T^{2} \)
79 \( 1 - 6.37T + 79T^{2} \)
83 \( 1 + (3.17 + 3.17i)T + 83iT^{2} \)
89 \( 1 + 13.9iT - 89T^{2} \)
97 \( 1 + 13.9T + 97T^{2} \)
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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

−8.999237522075532778171469589251, −8.298334266879883141284126998484, −7.71993968936312658767854226353, −6.57082560945802054013755461057, −5.95155129393610083589906700773, −5.05292828082703495405954853049, −4.70971954711268768720265878171, −3.03982147672467441709170180854, −2.60505906145708736132013917269, −1.40220632364660249863071433843, 0.37481612425118163380153327397, 1.68510949336885167312800137444, 2.79966621377201694298874276816, 3.84180241061113332980614855019, 4.46296136145939568785877280569, 5.41896221672136334243575204072, 6.53331643921366324600044324713, 6.81813968911279447435439575933, 7.896734670824998888461378547987, 8.245166689220227011229517272401

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