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Properties

Label 2-2880-15.2-c1-0-31
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 − 5.65i·11-s + (−3 + 3i)13-s − 4i·19-s + (2.82 + 2.82i)23-s + (3.99 + 3i)25-s + 1.41·29-s + 8·31-s + (−7 − 7i)37-s + 1.41i·41-s + (4 − 4i)43-s + (2.82 − 2.82i)47-s − 7i·49-s + (−8.48 − 8.48i)53-s + (4.00 − 12i)55-s + ⋯
L(s)  = 1  + (0.948 + 0.316i)5-s − 1.70i·11-s + (−0.832 + 0.832i)13-s − 0.917i·19-s + (0.589 + 0.589i)23-s + (0.799 + 0.600i)25-s + 0.262·29-s + 1.43·31-s + (−1.15 − 1.15i)37-s + 0.220i·41-s + (0.609 − 0.609i)43-s + (0.412 − 0.412i)47-s − i·49-s + (−1.16 − 1.16i)53-s + (0.539 − 1.61i)55-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: $\chi_{2880} (2177, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.073724570\)
\(L(\frac12)\) \(\approx\) \(2.073724570\)
\(L(\frac{3}{2})\) 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 \( 1 + 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 + (3 - 3i)T - 13iT^{2} \)
17 \( 1 - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-2.82 - 2.82i)T + 23iT^{2} \)
29 \( 1 - 1.41T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (7 + 7i)T + 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (-4 + 4i)T - 43iT^{2} \)
47 \( 1 + (-2.82 + 2.82i)T - 47iT^{2} \)
53 \( 1 + (8.48 + 8.48i)T + 53iT^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 12T + 61T^{2} \)
67 \( 1 + (8 + 8i)T + 67iT^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + (3 - 3i)T - 73iT^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 - 7.07T + 89T^{2} \)
97 \( 1 + (-5 - 5i)T + 97iT^{2} \)
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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

−8.888527432638473562066579766744, −8.008710107123627517069547091010, −6.88159476979488878062936829557, −6.57546892835803137482622581724, −5.51736631244696594508427360856, −5.06021930871127566272128805278, −3.80056380607333870835486631893, −2.88969042146603586400369992828, −2.08632734781656558586723080781, −0.72061640829780883741166664823, 1.17106132764155026315180304810, 2.21314163964743257704286779755, 2.97290567139225049658401127234, 4.46419195160507331728356325473, 4.87351948303845375843259535202, 5.77208687405961952135496594669, 6.58511481783794241201278086620, 7.33157220739800511448933551102, 8.097916979141637265940164421710, 8.927802309684643794176871482816

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