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Properties

Label 2-675-15.14-c2-0-30
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  + 1.57·2-s − 1.52·4-s + 2.52i·7-s − 8.69·8-s − 1.57i·11-s − 12.0i·13-s + 3.96i·14-s − 7.60·16-s + 24.4·17-s + 22.5·19-s − 2.47i·22-s + 22.1·23-s − 18.9i·26-s − 3.83i·28-s − 47.3i·29-s + ⋯
L(s)  = 1  + 0.787·2-s − 0.380·4-s + 0.360i·7-s − 1.08·8-s − 0.143i·11-s − 0.926i·13-s + 0.283i·14-s − 0.475·16-s + 1.43·17-s + 1.18·19-s − 0.112i·22-s + 0.961·23-s − 0.729i·26-s − 0.136i·28-s − 1.63i·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

\(L(\frac{3}{2})\) \(\approx\) \(2.277047165\)
\(L(\frac12)\) \(\approx\) \(2.277047165\)
\(L(2)\) 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 \( 1 - 1.57T + 4T^{2} \)
7 \( 1 - 2.52iT - 49T^{2} \)
11 \( 1 + 1.57iT - 121T^{2} \)
13 \( 1 + 12.0iT - 169T^{2} \)
17 \( 1 - 24.4T + 289T^{2} \)
19 \( 1 - 22.5T + 361T^{2} \)
23 \( 1 - 22.1T + 529T^{2} \)
29 \( 1 + 47.3iT - 841T^{2} \)
31 \( 1 - 23.5T + 961T^{2} \)
37 \( 1 - 21.5iT - 1.36e3T^{2} \)
41 \( 1 - 65.4iT - 1.68e3T^{2} \)
43 \( 1 + 58.7iT - 1.84e3T^{2} \)
47 \( 1 + 57.5T + 2.20e3T^{2} \)
53 \( 1 + 19.7T + 2.80e3T^{2} \)
59 \( 1 + 37.9iT - 3.48e3T^{2} \)
61 \( 1 - 46.6T + 3.72e3T^{2} \)
67 \( 1 + 120. iT - 4.48e3T^{2} \)
71 \( 1 - 48.0iT - 5.04e3T^{2} \)
73 \( 1 + 10.3iT - 5.32e3T^{2} \)
79 \( 1 - 101.T + 6.24e3T^{2} \)
83 \( 1 + 43.2T + 6.88e3T^{2} \)
89 \( 1 - 118. iT - 7.92e3T^{2} \)
97 \( 1 - 39.8iT - 9.40e3T^{2} \)
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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

−10.02126718899659217080067720039, −9.559046164895036817883013476144, −8.392370402141927349980206248292, −7.72655028214137279979836109021, −6.35502988479479376652975421434, −5.49368350509844906317641743157, −4.89009316279394039691671686540, −3.55424886211183817500975374507, −2.85261592469280385086900935866, −0.822071795249492711874264135118, 1.12547139943543959613628746588, 2.98876312893321038418813154622, 3.82740891961407633740594872788, 4.90343583250163062495510807571, 5.56401829970419262476141193966, 6.74574329095029342544366785800, 7.59324045670337205187097878613, 8.773597327958953579151210074416, 9.469330751140458841221161974445, 10.31012817293447169090127776325

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