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Properties

Label 2-2880-15.8-c1-0-20
Degree $2$
Conductor $2880$
Sign $0.990 + 0.139i$
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  + (−1.91 − 1.15i)5-s + (1.70 − 1.70i)7-s + 0.892i·11-s + (2.70 + 2.70i)13-s + (−1.65 − 1.65i)17-s + 7.41i·19-s + (6.13 − 6.13i)23-s + (2.34 + 4.41i)25-s + 0.521·29-s − 5.26·31-s + (−5.24 + 1.30i)35-s + (−1.78 + 1.78i)37-s + 0.110i·41-s + (1.26 + 1.26i)43-s + (6.77 + 6.77i)47-s + ⋯
L(s)  = 1  + (−0.856 − 0.515i)5-s + (0.646 − 0.646i)7-s + 0.269i·11-s + (0.751 + 0.751i)13-s + (−0.401 − 0.401i)17-s + 1.70i·19-s + (1.27 − 1.27i)23-s + (0.468 + 0.883i)25-s + 0.0969·29-s − 0.945·31-s + (−0.886 + 0.220i)35-s + (−0.293 + 0.293i)37-s + 0.0173i·41-s + (0.192 + 0.192i)43-s + (0.987 + 0.987i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.990 + 0.139i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.990 + 0.139i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.682275298\)
\(L(\frac12)\) \(\approx\) \(1.682275298\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.91 + 1.15i)T \)
good7 \( 1 + (-1.70 + 1.70i)T - 7iT^{2} \)
11 \( 1 - 0.892iT - 11T^{2} \)
13 \( 1 + (-2.70 - 2.70i)T + 13iT^{2} \)
17 \( 1 + (1.65 + 1.65i)T + 17iT^{2} \)
19 \( 1 - 7.41iT - 19T^{2} \)
23 \( 1 + (-6.13 + 6.13i)T - 23iT^{2} \)
29 \( 1 - 0.521T + 29T^{2} \)
31 \( 1 + 5.26T + 31T^{2} \)
37 \( 1 + (1.78 - 1.78i)T - 37iT^{2} \)
41 \( 1 - 0.110iT - 41T^{2} \)
43 \( 1 + (-1.26 - 1.26i)T + 43iT^{2} \)
47 \( 1 + (-6.77 - 6.77i)T + 47iT^{2} \)
53 \( 1 + (-4.07 + 4.07i)T - 53iT^{2} \)
59 \( 1 + 10.5T + 59T^{2} \)
61 \( 1 - 12.0T + 61T^{2} \)
67 \( 1 + (-5.26 + 5.26i)T - 67iT^{2} \)
71 \( 1 - 3.05iT - 71T^{2} \)
73 \( 1 + (-7.34 - 7.34i)T + 73iT^{2} \)
79 \( 1 + 13.2iT - 79T^{2} \)
83 \( 1 + (6.54 - 6.54i)T - 83iT^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + (-8.60 + 8.60i)T - 97iT^{2} \)
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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

−8.659255251260897657778943525945, −8.045727267417419202736331731478, −7.31714307452218595532475110480, −6.65414819944900741790422032419, −5.59171349693954683441799730916, −4.61225857476185891394674253386, −4.17892507083949511411178418060, −3.30036413682171695027403799272, −1.84851303185708790000077461775, −0.864014007397062373734065823046, 0.77866167873510901697282899600, 2.23813545977118386335821464596, 3.18377691480247811973865520811, 3.91369591842572272232178675963, 5.02006003100941146561488917453, 5.57901646149329150432250906536, 6.67502059191644170616242030314, 7.29481040940747506061566022504, 8.056744533978901553446910771659, 8.799290738865028809795223524569

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