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Properties

Label 2-4560-76.75-c1-0-50
Degree $2$
Conductor $4560$
Sign $0.920 + 0.390i$
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 + 0.855i·7-s + 9-s + 2.02i·11-s − 6.27i·13-s − 15-s + 4.01·17-s + (0.531 + 4.32i)19-s + 0.855i·21-s − 2.66i·23-s + 25-s + 27-s − 2.52i·29-s − 2.57·31-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 0.323i·7-s + 0.333·9-s + 0.612i·11-s − 1.73i·13-s − 0.258·15-s + 0.972·17-s + (0.121 + 0.992i)19-s + 0.186i·21-s − 0.554i·23-s + 0.200·25-s + 0.192·27-s − 0.469i·29-s − 0.463·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $0.920 + 0.390i$
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: \((2,\ 4560,\ (\ :1/2),\ 0.920 + 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.206127188\)
\(L(\frac12)\) \(\approx\) \(2.206127188\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.531 - 4.32i)T \)
good7 \( 1 - 0.855iT - 7T^{2} \)
11 \( 1 - 2.02iT - 11T^{2} \)
13 \( 1 + 6.27iT - 13T^{2} \)
17 \( 1 - 4.01T + 17T^{2} \)
23 \( 1 + 2.66iT - 23T^{2} \)
29 \( 1 + 2.52iT - 29T^{2} \)
31 \( 1 + 2.57T + 31T^{2} \)
37 \( 1 + 1.25iT - 37T^{2} \)
41 \( 1 - 0.709iT - 41T^{2} \)
43 \( 1 + 2.20iT - 43T^{2} \)
47 \( 1 - 1.98iT - 47T^{2} \)
53 \( 1 + 3.05iT - 53T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 2.23T + 61T^{2} \)
67 \( 1 + 4.14T + 67T^{2} \)
71 \( 1 - 5.04T + 71T^{2} \)
73 \( 1 + 9.52T + 73T^{2} \)
79 \( 1 + 4.17T + 79T^{2} \)
83 \( 1 + 12.4iT - 83T^{2} \)
89 \( 1 - 14.1iT - 89T^{2} \)
97 \( 1 + 5.31iT - 97T^{2} \)
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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

−8.223805418518580181158338357094, −7.66140255042792099602180826594, −7.11338237090817922120778467840, −5.92314973094976634535198602837, −5.43745860580735516261229344484, −4.45198031782434238229292376510, −3.57349984753150967918386652406, −2.96237365999683087029018991654, −1.97137209307088541502278062890, −0.71326388615507887075724614456, 0.942180415543340960216242941502, 2.01684160722093923568521206902, 3.08486892863959632730411983511, 3.78688043477905057995360546074, 4.48312329360917087220394869925, 5.35524081301745957111305007017, 6.33627097958148079586230201521, 7.16560912793893158188669779768, 7.48924723891002554282126486317, 8.531199563794987777240113549915

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