Properties

Label 2-2880-48.35-c1-0-18
Degree $2$
Conductor $2880$
Sign $0.830 - 0.556i$
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 + 4.13·7-s + (−1.99 + 1.99i)11-s + (0.516 + 0.516i)13-s − 3.26i·17-s + (3.73 − 3.73i)19-s + 7.20i·23-s + 1.00i·25-s + (−1.21 + 1.21i)29-s + 9.55i·31-s + (2.92 + 2.92i)35-s + (6.37 − 6.37i)37-s + 7.26·41-s + (−0.127 − 0.127i)43-s + 8.87·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 1.56·7-s + (−0.601 + 0.601i)11-s + (0.143 + 0.143i)13-s − 0.790i·17-s + (0.856 − 0.856i)19-s + 1.50i·23-s + 0.200i·25-s + (−0.226 + 0.226i)29-s + 1.71i·31-s + (0.494 + 0.494i)35-s + (1.04 − 1.04i)37-s + 1.13·41-s + (−0.0194 − 0.0194i)43-s + 1.29·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.830 - 0.556i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.830 - 0.556i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.392093638\)
\(L(\frac12)\) \(\approx\) \(2.392093638\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.13T + 7T^{2} \)
11 \( 1 + (1.99 - 1.99i)T - 11iT^{2} \)
13 \( 1 + (-0.516 - 0.516i)T + 13iT^{2} \)
17 \( 1 + 3.26iT - 17T^{2} \)
19 \( 1 + (-3.73 + 3.73i)T - 19iT^{2} \)
23 \( 1 - 7.20iT - 23T^{2} \)
29 \( 1 + (1.21 - 1.21i)T - 29iT^{2} \)
31 \( 1 - 9.55iT - 31T^{2} \)
37 \( 1 + (-6.37 + 6.37i)T - 37iT^{2} \)
41 \( 1 - 7.26T + 41T^{2} \)
43 \( 1 + (0.127 + 0.127i)T + 43iT^{2} \)
47 \( 1 - 8.87T + 47T^{2} \)
53 \( 1 + (9.72 + 9.72i)T + 53iT^{2} \)
59 \( 1 + (2.86 - 2.86i)T - 59iT^{2} \)
61 \( 1 + (4.64 + 4.64i)T + 61iT^{2} \)
67 \( 1 + (0.264 - 0.264i)T - 67iT^{2} \)
71 \( 1 - 1.88iT - 71T^{2} \)
73 \( 1 - 2.96iT - 73T^{2} \)
79 \( 1 - 6.08iT - 79T^{2} \)
83 \( 1 + (-4.98 - 4.98i)T + 83iT^{2} \)
89 \( 1 + 6.90T + 89T^{2} \)
97 \( 1 + 10.2T + 97T^{2} \)
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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

−8.986481968478415747796060517056, −7.78934668645172385160653983230, −7.55767159257048452356746218856, −6.73209733925611148941517202243, −5.42439908692135433140374518838, −5.16398108059111808167628824830, −4.26359429430471633836273223602, −3.06019399834101204153772225510, −2.13479674506682743508735540048, −1.17636046799506612100508511683, 0.894013105688607923671876978768, 1.90569390837324886772339875721, 2.87708134281703046930870761695, 4.21339477009998123359556718569, 4.70457820494087888790369637719, 5.81826284015829477282225140098, 6.02854120962818687388962771088, 7.55877753483240490008662620682, 7.960355634785419767930481940199, 8.518260771489726056211286359388

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