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Properties

Label 2-2880-16.13-c1-0-31
Degree $2$
Conductor $2880$
Sign $-0.627 + 0.778i$
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  + (−0.707 + 0.707i)5-s + 4.27i·7-s + (2.94 − 2.94i)11-s + (−4.05 − 4.05i)13-s − 0.160·17-s + (−4.32 − 4.32i)19-s + 8.40i·23-s − 1.00i·25-s + (1.78 + 1.78i)29-s − 7.17·31-s + (−3.02 − 3.02i)35-s + (−0.669 + 0.669i)37-s − 3.96i·41-s + (0.255 − 0.255i)43-s + 0.0752·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + 1.61i·7-s + (0.887 − 0.887i)11-s + (−1.12 − 1.12i)13-s − 0.0388·17-s + (−0.992 − 0.992i)19-s + 1.75i·23-s − 0.200i·25-s + (0.330 + 0.330i)29-s − 1.28·31-s + (−0.510 − 0.510i)35-s + (−0.110 + 0.110i)37-s − 0.619i·41-s + (0.0389 − 0.0389i)43-s + 0.0109·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.627 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.627 + 0.778i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.627 + 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3092956439\)
\(L(\frac12)\) \(\approx\) \(0.3092956439\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 - 0.707i)T \)
good7 \( 1 - 4.27iT - 7T^{2} \)
11 \( 1 + (-2.94 + 2.94i)T - 11iT^{2} \)
13 \( 1 + (4.05 + 4.05i)T + 13iT^{2} \)
17 \( 1 + 0.160T + 17T^{2} \)
19 \( 1 + (4.32 + 4.32i)T + 19iT^{2} \)
23 \( 1 - 8.40iT - 23T^{2} \)
29 \( 1 + (-1.78 - 1.78i)T + 29iT^{2} \)
31 \( 1 + 7.17T + 31T^{2} \)
37 \( 1 + (0.669 - 0.669i)T - 37iT^{2} \)
41 \( 1 + 3.96iT - 41T^{2} \)
43 \( 1 + (-0.255 + 0.255i)T - 43iT^{2} \)
47 \( 1 - 0.0752T + 47T^{2} \)
53 \( 1 + (-2.88 + 2.88i)T - 53iT^{2} \)
59 \( 1 + (5.63 - 5.63i)T - 59iT^{2} \)
61 \( 1 + (4.48 + 4.48i)T + 61iT^{2} \)
67 \( 1 + (-0.131 - 0.131i)T + 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 0.382iT - 73T^{2} \)
79 \( 1 + 15.3T + 79T^{2} \)
83 \( 1 + (5.54 + 5.54i)T + 83iT^{2} \)
89 \( 1 + 13.8iT - 89T^{2} \)
97 \( 1 - 10.8T + 97T^{2} \)
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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

−8.760839187319551928112824710325, −7.73648330265210786691042456383, −7.05522978336340148757285228492, −6.02937834172919323771511988452, −5.56172448797849422176968474192, −4.71456846454512889018128441679, −3.45280464802960777559760879267, −2.85294152544082677147581100140, −1.84006642333951002995344595577, −0.097386670731913891536259896519, 1.31104366266075723601257844602, 2.30275561216476527280384903701, 3.85396379244873760012946837249, 4.27913589975352370384116413945, 4.79732842062895517836030142444, 6.25192806693484735262067046240, 6.92476584239549369279858491325, 7.36850983992631514999821991247, 8.233149259301711485110126653388, 9.081358149088578665367638659075

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