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Properties

Label 2-2880-15.2-c1-0-28
Degree $2$
Conductor $2880$
Sign $0.927 - 0.374i$
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  + (1.58 − 1.58i)5-s + (3.23 + 3.23i)7-s + 4.57i·11-s + (4.23 − 4.23i)13-s + (−1.74 + 1.74i)17-s − 2.47i·19-s + (2.82 + 2.82i)23-s − 5.00i·25-s + 5.99·29-s − 1.52·31-s + 10.2·35-s + (−2.23 − 2.23i)37-s + 7.07i·41-s + (2.47 − 2.47i)43-s + (−1.74 + 1.74i)47-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (1.22 + 1.22i)7-s + 1.37i·11-s + (1.17 − 1.17i)13-s + (−0.423 + 0.423i)17-s − 0.567i·19-s + (0.589 + 0.589i)23-s − 1.00i·25-s + 1.11·29-s − 0.274·31-s + 1.72·35-s + (−0.367 − 0.367i)37-s + 1.10i·41-s + (0.376 − 0.376i)43-s + (−0.254 + 0.254i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.927 - 0.374i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.927 - 0.374i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.666763097\)
\(L(\frac12)\) \(\approx\) \(2.666763097\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.58 + 1.58i)T \)
good7 \( 1 + (-3.23 - 3.23i)T + 7iT^{2} \)
11 \( 1 - 4.57iT - 11T^{2} \)
13 \( 1 + (-4.23 + 4.23i)T - 13iT^{2} \)
17 \( 1 + (1.74 - 1.74i)T - 17iT^{2} \)
19 \( 1 + 2.47iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 - 5.99T + 29T^{2} \)
31 \( 1 + 1.52T + 31T^{2} \)
37 \( 1 + (2.23 + 2.23i)T + 37iT^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + (-2.47 + 2.47i)T - 43iT^{2} \)
47 \( 1 + (1.74 - 1.74i)T - 47iT^{2} \)
53 \( 1 + 53iT^{2} \)
59 \( 1 + 1.08T + 59T^{2} \)
61 \( 1 + 10.4T + 61T^{2} \)
67 \( 1 + (1.52 + 1.52i)T + 67iT^{2} \)
71 \( 1 - 12.6iT - 71T^{2} \)
73 \( 1 + (0.527 - 0.527i)T - 73iT^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 + (1.08 + 1.08i)T + 83iT^{2} \)
89 \( 1 + 0.746T + 89T^{2} \)
97 \( 1 + (-1 - i)T + 97iT^{2} \)
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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

−8.764874259339105649881395608615, −8.270555202214735282404498699415, −7.47214427570732125308608981975, −6.32711815460551964265398849694, −5.64863385903793265649609328887, −4.97289141760207339616510066404, −4.42196475887753710428075420294, −2.94357818585704041633019274230, −1.98893325814664009360162994780, −1.26029947369499175908028350319, 0.998830467297370562095106850191, 1.85841080690812148414982781687, 3.11938069031410447605846045971, 3.96282511379120459271907464414, 4.76816724698642741082243283041, 5.77084514340414840916062653070, 6.53504337298383930975491368929, 7.07134558260315477186104463196, 8.055512465498433981252261601751, 8.659047694880272368882720187204

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