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Properties

Label 2-4560-76.75-c1-0-10
Degree $2$
Conductor $4560$
Sign $-0.717 - 0.696i$
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 + 2.79i·7-s + 9-s + 5.01i·11-s − 3.47i·13-s − 15-s + 2.25·17-s + (−4.19 + 1.19i)19-s − 2.79i·21-s + 0.453i·23-s + 25-s − 27-s + 1.54i·29-s + 5.12·31-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1.05i·7-s + 0.333·9-s + 1.51i·11-s − 0.964i·13-s − 0.258·15-s + 0.547·17-s + (−0.961 + 0.273i)19-s − 0.608i·21-s + 0.0946i·23-s + 0.200·25-s − 0.192·27-s + 0.287i·29-s + 0.921·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.717 - 0.696i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4560\)    =    \(2^{4} \cdot 3 \cdot 5 \cdot 19\)
Sign: $-0.717 - 0.696i$
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: \((2,\ 4560,\ (\ :1/2),\ -0.717 - 0.696i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.160767817\)
\(L(\frac12)\) \(\approx\) \(1.160767817\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.19 - 1.19i)T \)
good7 \( 1 - 2.79iT - 7T^{2} \)
11 \( 1 - 5.01iT - 11T^{2} \)
13 \( 1 + 3.47iT - 13T^{2} \)
17 \( 1 - 2.25T + 17T^{2} \)
23 \( 1 - 0.453iT - 23T^{2} \)
29 \( 1 - 1.54iT - 29T^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 - 4.77iT - 37T^{2} \)
41 \( 1 - 2.33iT - 41T^{2} \)
43 \( 1 + 1.78iT - 43T^{2} \)
47 \( 1 - 8.48iT - 47T^{2} \)
53 \( 1 - 3.01iT - 53T^{2} \)
59 \( 1 - 1.34T + 59T^{2} \)
61 \( 1 + 6.21T + 61T^{2} \)
67 \( 1 - 3.58T + 67T^{2} \)
71 \( 1 - 5.38T + 71T^{2} \)
73 \( 1 + 11.7T + 73T^{2} \)
79 \( 1 + 2.18T + 79T^{2} \)
83 \( 1 + 9.49iT - 83T^{2} \)
89 \( 1 + 3.89iT - 89T^{2} \)
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

−8.581894295612401776523250581510, −7.87132542377824697357406699564, −7.09030165030278119583083199802, −6.24075995056523416973836386937, −5.74765258538246992253915576568, −4.96025101170157590365499835857, −4.36759982917272596469689505959, −3.06578230925977848288116506593, −2.26382975131538263798396161922, −1.32645054045206644075754557124, 0.36631579523645178015097081061, 1.31272074718391264656227916979, 2.49940273745071890285525320214, 3.65724612620579257401731477777, 4.23967035465594603501117699888, 5.16405883254397346094199448084, 5.96044980009043078805880182001, 6.54586691350548538979312963197, 7.14990321880876968328165875069, 8.088733810294591026846956015207

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