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Properties

Label 2-2880-15.2-c1-0-6
Degree $2$
Conductor $2880$
Sign $-0.241 - 0.970i$
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  + (1.34 + 1.78i)5-s + (−2.90 − 2.90i)7-s − 2.15i·11-s + (−1.90 + 1.90i)13-s + (0.974 − 0.974i)17-s + 1.80i·19-s + (0.879 + 0.879i)23-s + (−1.37 + 4.80i)25-s − 0.743·29-s − 7.05·31-s + (1.27 − 9.09i)35-s + (8.33 + 8.33i)37-s + 7.67i·41-s + (3.05 − 3.05i)43-s + (−7.54 + 7.54i)47-s + ⋯
L(s)  = 1  + (0.601 + 0.798i)5-s + (−1.09 − 1.09i)7-s − 0.650i·11-s + (−0.527 + 0.527i)13-s + (0.236 − 0.236i)17-s + 0.414i·19-s + (0.183 + 0.183i)23-s + (−0.275 + 0.961i)25-s − 0.137·29-s − 1.26·31-s + (0.215 − 1.53i)35-s + (1.36 + 1.36i)37-s + 1.19i·41-s + (0.465 − 0.465i)43-s + (−1.09 + 1.09i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.241 - 0.970i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.241 - 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.036663072\)
\(L(\frac12)\) \(\approx\) \(1.036663072\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 - 1.78i)T \)
good7 \( 1 + (2.90 + 2.90i)T + 7iT^{2} \)
11 \( 1 + 2.15iT - 11T^{2} \)
13 \( 1 + (1.90 - 1.90i)T - 13iT^{2} \)
17 \( 1 + (-0.974 + 0.974i)T - 17iT^{2} \)
19 \( 1 - 1.80iT - 19T^{2} \)
23 \( 1 + (-0.879 - 0.879i)T + 23iT^{2} \)
29 \( 1 + 0.743T + 29T^{2} \)
31 \( 1 + 7.05T + 31T^{2} \)
37 \( 1 + (-8.33 - 8.33i)T + 37iT^{2} \)
41 \( 1 - 7.67iT - 41T^{2} \)
43 \( 1 + (-3.05 + 3.05i)T - 43iT^{2} \)
47 \( 1 + (7.54 - 7.54i)T - 47iT^{2} \)
53 \( 1 + (5.08 + 5.08i)T + 53iT^{2} \)
59 \( 1 - 14.2T + 59T^{2} \)
61 \( 1 + 4.56T + 61T^{2} \)
67 \( 1 + (-7.05 - 7.05i)T + 67iT^{2} \)
71 \( 1 - 12.5iT - 71T^{2} \)
73 \( 1 + (-3.62 + 3.62i)T - 73iT^{2} \)
79 \( 1 - 15.0iT - 79T^{2} \)
83 \( 1 + (7.81 + 7.81i)T + 83iT^{2} \)
89 \( 1 + 7.40T + 89T^{2} \)
97 \( 1 + (-6.67 - 6.67i)T + 97iT^{2} \)
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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

−9.265648266099552975841998361005, −8.104025548701619395053037379464, −7.31354864357869471693898040588, −6.69104574713720738219897805631, −6.16209356594272872922285615244, −5.23610042617526357486282084178, −4.06742063883629130508783291563, −3.34056469934002607960522318620, −2.58060455316754494111709944550, −1.20401920139198979487071754777, 0.33635345645803958554736127058, 1.94177017996433306792390728558, 2.64810936833266384445415146235, 3.73266439312221732419025838182, 4.84106587591133258050493581577, 5.54272087994574702793353371604, 6.08069837577651695069268344380, 6.99515226671328905062229957286, 7.84677412715475833039121032355, 8.805901072627034406383740032503

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