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Properties

Label 2-2880-15.2-c1-0-42
Degree $2$
Conductor $2880$
Sign $-0.374 + 0.927i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.374 + 0.927i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.330414804\)
\(L(\frac12)\) \(\approx\) \(1.330414804\)
\(L(\frac{3}{2})\) 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

−8.719357494008601468322324279652, −7.54816663746624371908149572841, −7.08477114355537582926270290965, −6.12278754483400874534532985928, −5.71703836849946035836658253437, −4.63638950124389487792952891275, −3.57372898888982727741412619583, −2.89561162214218124473398188214, −1.76089066474113702984598841071, −0.40054940699197612981273884778, 1.47149077753794754130507485925, 2.33874463484166246641401170656, 3.27584634168450388071284310521, 4.36015747967678102555755329124, 5.40726479872284043479699904564, 5.82374201283478998469742107460, 6.60249109322810522229280460562, 7.51194143427941230583572617004, 8.305927515184432756593090365869, 9.279924779301045916000871546562

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