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Properties

Label 2-2880-15.8-c1-0-33
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: $\chi_{2880} (2753, \cdot )$
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

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

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