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Properties

Label 2-675-15.14-c2-0-44
Degree $2$
Conductor $675$
Sign $-0.894 - 0.447i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.69·2-s + 9.64·4-s − 9.20i·7-s − 20.8·8-s − 15.5i·11-s − 20.6i·13-s + 34.0i·14-s + 38.4·16-s − 22.4·17-s − 2.33·19-s + 57.4i·22-s − 19.3·23-s + 76.0i·26-s − 88.8i·28-s − 2.92i·29-s + ⋯
L(s)  = 1  − 1.84·2-s + 2.41·4-s − 1.31i·7-s − 2.60·8-s − 1.41i·11-s − 1.58i·13-s + 2.42i·14-s + 2.40·16-s − 1.32·17-s − 0.122·19-s + 2.61i·22-s − 0.839·23-s + 2.92i·26-s − 3.17i·28-s − 0.100i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (674, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ -0.894 - 0.447i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.2629724752\)
\(L(\frac12)\) \(\approx\) \(0.2629724752\)
\(L(2)\) 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 \( 1 + 3.69T + 4T^{2} \)
7 \( 1 + 9.20iT - 49T^{2} \)
11 \( 1 + 15.5iT - 121T^{2} \)
13 \( 1 + 20.6iT - 169T^{2} \)
17 \( 1 + 22.4T + 289T^{2} \)
19 \( 1 + 2.33T + 361T^{2} \)
23 \( 1 + 19.3T + 529T^{2} \)
29 \( 1 + 2.92iT - 841T^{2} \)
31 \( 1 + 28.1T + 961T^{2} \)
37 \( 1 - 65.4iT - 1.36e3T^{2} \)
41 \( 1 + 7.48iT - 1.68e3T^{2} \)
43 \( 1 + 11.9iT - 1.84e3T^{2} \)
47 \( 1 - 50.5T + 2.20e3T^{2} \)
53 \( 1 - 80.8T + 2.80e3T^{2} \)
59 \( 1 - 9.46iT - 3.48e3T^{2} \)
61 \( 1 - 52.7T + 3.72e3T^{2} \)
67 \( 1 - 65.2iT - 4.48e3T^{2} \)
71 \( 1 + 0.942iT - 5.04e3T^{2} \)
73 \( 1 - 12.8iT - 5.32e3T^{2} \)
79 \( 1 + 68.4T + 6.24e3T^{2} \)
83 \( 1 + 38.9T + 6.88e3T^{2} \)
89 \( 1 - 63.7iT - 7.92e3T^{2} \)
97 \( 1 - 186. iT - 9.40e3T^{2} \)
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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

−9.929351247795320507943760563156, −8.755295031644372425967353114680, −8.251457026717938326662797521830, −7.44425717374020996253657304315, −6.63729093320680882740141943432, −5.65200018547171119941555100074, −3.83194963886353548978936833711, −2.63583948159570125184885029050, −1.05078964710140886132073665202, −0.19579918079945404992734294801, 2.00690884313792020103700423565, 2.20470461491553139086681662885, 4.28754924005453837787360866059, 5.79967735347598620333941412660, 6.80064964835937895232872834857, 7.35870475778525292342240284308, 8.559811190424339770980331616630, 9.121629063154706741657186171634, 9.561644566335614121456279639096, 10.57758584375985371848979023393

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