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Properties

Label 2-2880-15.8-c1-0-11
Degree $2$
Conductor $2880$
Sign $0.749 - 0.662i$
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  + (−2.12 + 0.707i)5-s + (−2 + 2i)7-s − 2.82i·11-s + (−1 − i)13-s + (−2.82 − 2.82i)17-s + (2.82 − 2.82i)23-s + (3.99 − 3i)25-s + 4.24·29-s − 4·31-s + (2.82 − 5.65i)35-s + (−1 + i)37-s − 1.41i·41-s + (8 + 8i)43-s + (5.65 + 5.65i)47-s − i·49-s + ⋯
L(s)  = 1  + (−0.948 + 0.316i)5-s + (−0.755 + 0.755i)7-s − 0.852i·11-s + (−0.277 − 0.277i)13-s + (−0.685 − 0.685i)17-s + (0.589 − 0.589i)23-s + (0.799 − 0.600i)25-s + 0.787·29-s − 0.718·31-s + (0.478 − 0.956i)35-s + (−0.164 + 0.164i)37-s − 0.220i·41-s + (1.21 + 1.21i)43-s + (0.825 + 0.825i)47-s − 0.142i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.749 - 0.662i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.007703692\)
\(L(\frac12)\) \(\approx\) \(1.007703692\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.12 - 0.707i)T \)
good7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 + 2.82iT - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (2.82 + 2.82i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-2.82 + 2.82i)T - 23iT^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + (-8 - 8i)T + 43iT^{2} \)
47 \( 1 + (-5.65 - 5.65i)T + 47iT^{2} \)
53 \( 1 + (2.82 - 2.82i)T - 53iT^{2} \)
59 \( 1 - 8.48T + 59T^{2} \)
61 \( 1 + 8T + 61T^{2} \)
67 \( 1 + (4 - 4i)T - 67iT^{2} \)
71 \( 1 + 5.65iT - 71T^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 - 12iT - 79T^{2} \)
83 \( 1 + (2.82 - 2.82i)T - 83iT^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (11 - 11i)T - 97iT^{2} \)
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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

−8.917334663829180579626180239568, −8.122763503714788041473088529865, −7.35634302829199093069867005088, −6.58238672404995185469779842267, −5.92283266947863563213741671386, −4.93745100804135427128760198089, −4.07685275620115631375384517506, −3.02044978432829138264785772496, −2.62847859470532680076236861483, −0.72120797507231385869767833994, 0.50249896278038551578087590942, 1.90342993093298606850859195284, 3.19848036589662489838252028688, 4.00944903880994180949782369336, 4.55173098515000557837842781592, 5.54797300191368730248724988189, 6.71679230052877212874048025516, 7.14243137093171452963192178043, 7.79401199541200593274476457820, 8.777329877028252432952771980782

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