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Properties

Label 2-2880-48.35-c1-0-13
Degree $2$
Conductor $2880$
Sign $0.626 - 0.779i$
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 + 3.01·7-s + (2.52 − 2.52i)11-s + (0.848 + 0.848i)13-s + 8.11i·17-s + (−1.76 + 1.76i)19-s + 8.68i·23-s + 1.00i·25-s + (−5.71 + 5.71i)29-s − 1.57i·31-s + (−2.13 − 2.13i)35-s + (4.56 − 4.56i)37-s − 11.3·41-s + (4.33 + 4.33i)43-s + 7.46·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s + 1.13·7-s + (0.759 − 0.759i)11-s + (0.235 + 0.235i)13-s + 1.96i·17-s + (−0.405 + 0.405i)19-s + 1.81i·23-s + 0.200i·25-s + (−1.06 + 1.06i)29-s − 0.282i·31-s + (−0.360 − 0.360i)35-s + (0.749 − 0.749i)37-s − 1.77·41-s + (0.661 + 0.661i)43-s + 1.08·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.626 - 0.779i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.626 - 0.779i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.895092614\)
\(L(\frac12)\) \(\approx\) \(1.895092614\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.01T + 7T^{2} \)
11 \( 1 + (-2.52 + 2.52i)T - 11iT^{2} \)
13 \( 1 + (-0.848 - 0.848i)T + 13iT^{2} \)
17 \( 1 - 8.11iT - 17T^{2} \)
19 \( 1 + (1.76 - 1.76i)T - 19iT^{2} \)
23 \( 1 - 8.68iT - 23T^{2} \)
29 \( 1 + (5.71 - 5.71i)T - 29iT^{2} \)
31 \( 1 + 1.57iT - 31T^{2} \)
37 \( 1 + (-4.56 + 4.56i)T - 37iT^{2} \)
41 \( 1 + 11.3T + 41T^{2} \)
43 \( 1 + (-4.33 - 4.33i)T + 43iT^{2} \)
47 \( 1 - 7.46T + 47T^{2} \)
53 \( 1 + (4.02 + 4.02i)T + 53iT^{2} \)
59 \( 1 + (-2.67 + 2.67i)T - 59iT^{2} \)
61 \( 1 + (-5.84 - 5.84i)T + 61iT^{2} \)
67 \( 1 + (-8.64 + 8.64i)T - 67iT^{2} \)
71 \( 1 + 6.90iT - 71T^{2} \)
73 \( 1 + 2.42iT - 73T^{2} \)
79 \( 1 - 12.0iT - 79T^{2} \)
83 \( 1 + (-4.26 - 4.26i)T + 83iT^{2} \)
89 \( 1 - 3.00T + 89T^{2} \)
97 \( 1 + 7.39T + 97T^{2} \)
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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

−8.767425232935360298259284661514, −8.137346717533587827393340839041, −7.61694719051109504260545754898, −6.52789968835216410403871173922, −5.76262838354674156672133534965, −5.07754308430838443021145207975, −3.84078292962440275083372525427, −3.71296648675096586030133354511, −1.89433054914076717151195389143, −1.28791496640206122940186423180, 0.65213459292317691519932274289, 2.00825920696461816397478990484, 2.83553480826738929501738145694, 4.13947348143723194228004922695, 4.64073146028643977201594357454, 5.45427362168826431039614219138, 6.62763625025357961680803478768, 7.11122314330079196798301130266, 7.914174203947566355290600497560, 8.608241737297473980191601449256

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