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Properties

Label 2-2880-60.59-c1-0-38
Degree $2$
Conductor $2880$
Sign $-0.805 + 0.592i$
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 − 6i·13-s + 4.24·17-s + (3.99 + 3i)25-s + 9.89i·29-s − 12i·37-s + 1.41i·41-s − 7·49-s − 12.7·53-s − 10·61-s + (−4.24 + 12.7i)65-s − 6i·73-s + (−8.99 − 3i)85-s − 18.3i·89-s − 18i·97-s + ⋯
L(s)  = 1  + (−0.948 − 0.316i)5-s − 1.66i·13-s + 1.02·17-s + (0.799 + 0.600i)25-s + 1.83i·29-s − 1.97i·37-s + 0.220i·41-s − 49-s − 1.74·53-s − 1.28·61-s + (−0.526 + 1.57i)65-s − 0.702i·73-s + (−0.976 − 0.325i)85-s − 1.94i·89-s − 1.82i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.805 + 0.592i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7474659368\)
\(L(\frac12)\) \(\approx\) \(0.7474659368\)
\(L(\frac{3}{2})\) 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 \( 1 + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 9.89iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 18.3iT - 89T^{2} \)
97 \( 1 + 18iT - 97T^{2} \)
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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

−8.385807338703391967822168701017, −7.66923545414607436186273501987, −7.28653076028586121912395992117, −6.06546148494935167464680147882, −5.31939642924819921024317444417, −4.62503756267678159935323704540, −3.44829073487706481654802646732, −3.08063301660503382020800422036, −1.42939919657847700598358526782, −0.25854935414654121881177736660, 1.34541006016027690662371859658, 2.60226033821790172229266870186, 3.57748127449401638057062749088, 4.30779398838333144128922440316, 5.03048874813687618606417851831, 6.32862065577842436880248203245, 6.68120351976110322304334509914, 7.80721438203646679008323079049, 8.031955246522057943637721914563, 9.122002249798104104693239322280

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