Properties

Label 2-675-5.4-c1-0-20
Degree $2$
Conductor $675$
Sign $-0.894 - 0.447i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
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  − 2.30i·2-s − 3.30·4-s − 2.60i·7-s + 3.00i·8-s + 4.60·11-s − 6.60i·13-s − 6·14-s + 0.302·16-s + 1.60i·17-s − 3.60·19-s − 10.6i·22-s + 3i·23-s − 15.2·26-s + 8.60i·28-s − 1.39·29-s + ⋯
L(s)  = 1  − 1.62i·2-s − 1.65·4-s − 0.984i·7-s + 1.06i·8-s + 1.38·11-s − 1.83i·13-s − 1.60·14-s + 0.0756·16-s + 0.389i·17-s − 0.827·19-s − 2.26i·22-s + 0.625i·23-s − 2.98·26-s + 1.62i·28-s − 0.258·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282551 + 1.19690i\)
\(L(\frac12)\) \(\approx\) \(0.282551 + 1.19690i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + 2.30iT - 2T^{2} \)
7 \( 1 + 2.60iT - 7T^{2} \)
11 \( 1 - 4.60T + 11T^{2} \)
13 \( 1 + 6.60iT - 13T^{2} \)
17 \( 1 - 1.60iT - 17T^{2} \)
19 \( 1 + 3.60T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 1.39T + 29T^{2} \)
31 \( 1 + 5.60T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + 4.60T + 41T^{2} \)
43 \( 1 + 0.605iT - 43T^{2} \)
47 \( 1 + 9.21iT - 47T^{2} \)
53 \( 1 + 1.60iT - 53T^{2} \)
59 \( 1 - 1.39T + 59T^{2} \)
61 \( 1 + 4.21T + 61T^{2} \)
67 \( 1 + 0.788iT - 67T^{2} \)
71 \( 1 - 7.39T + 71T^{2} \)
73 \( 1 + 12.6iT - 73T^{2} \)
79 \( 1 - 11.6T + 79T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−10.37604016819879575772007519633, −9.459321573542760774861707261327, −8.596380389409863398849716600832, −7.52490253636756123154242469781, −6.39167326965225354527598354564, −5.05903544583601989574463104512, −3.84459755256625864013118812654, −3.42031498201743255397883523954, −1.86710467359163906338031227725, −0.67692972901026458455109863076, 2.04575594871964318394423956309, 3.98527890432724158685654259874, 4.80068737351865729137662196521, 5.96113950496678308390715207903, 6.55561364681611844453535830232, 7.21878542517889778180208383321, 8.454508847258919976208723225796, 9.066079287695574933004183147155, 9.466039048913983155427190265871, 11.12828188872104221444779500042

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