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Properties

Label 2-4560-76.75-c1-0-77
Degree $2$
Conductor $4560$
Sign $-0.738 + 0.674i$
Analytic cond. $36.4117$
Root an. cond. $6.03421$
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  + 3-s + 5-s − 4.69i·7-s + 9-s + 1.75i·11-s − 4.69i·13-s + 15-s − 4.86·17-s + (−3.21 + 2.93i)19-s − 4.69i·21-s − 4.44i·23-s + 25-s + 27-s − 7.14i·29-s + 6.21·31-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s − 1.77i·7-s + 0.333·9-s + 0.528i·11-s − 1.30i·13-s + 0.258·15-s − 1.17·17-s + (−0.738 + 0.674i)19-s − 1.02i·21-s − 0.927i·23-s + 0.200·25-s + 0.192·27-s − 1.32i·29-s + 1.11·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.738 + 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.738 + 0.674i$
Analytic conductor: \(36.4117\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4560} (2431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4560,\ (\ :1/2),\ -0.738 + 0.674i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.750278021\)
\(L(\frac12)\) \(\approx\) \(1.750278021\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
19 \( 1 + (3.21 - 2.93i)T \)
good7 \( 1 + 4.69iT - 7T^{2} \)
11 \( 1 - 1.75iT - 11T^{2} \)
13 \( 1 + 4.69iT - 13T^{2} \)
17 \( 1 + 4.86T + 17T^{2} \)
23 \( 1 + 4.44iT - 23T^{2} \)
29 \( 1 + 7.14iT - 29T^{2} \)
31 \( 1 - 6.21T + 31T^{2} \)
37 \( 1 - 6.20iT - 37T^{2} \)
41 \( 1 - 4.76iT - 41T^{2} \)
43 \( 1 - 6.20iT - 43T^{2} \)
47 \( 1 + 7.46iT - 47T^{2} \)
53 \( 1 + 4.44iT - 53T^{2} \)
59 \( 1 + 11.1T + 59T^{2} \)
61 \( 1 + 15.3T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 11.1T + 71T^{2} \)
73 \( 1 + 9.79T + 73T^{2} \)
79 \( 1 + 1.35T + 79T^{2} \)
83 \( 1 + 10.8iT - 83T^{2} \)
89 \( 1 + 15.5iT - 89T^{2} \)
97 \( 1 + 1.67iT - 97T^{2} \)
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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

−8.074792641729398500793876197770, −7.43256985774748873727807329982, −6.58286810865698844539092934679, −6.14403602131438853898132731067, −4.63677897449071317424722089785, −4.47702916262619008042661091707, −3.43906416586518157199255090182, −2.57528048158978341751358870108, −1.53854868221792931938840250557, −0.40672576343319535643677622047, 1.65144556158986412850285318518, 2.33078958863853214793551429698, 2.97425888457489037795315764132, 4.11966391760457706667860301044, 4.93446594091518043708701445385, 5.72348459695912384377189709745, 6.44698413713632255453921947787, 7.00782899378138488167873414854, 8.145214480922468621313800424959, 8.810569881008000487831901529091

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