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Properties

Label 2-675-15.14-c2-0-23
Degree $2$
Conductor $675$
Sign $0.447 - 0.894i$
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  + 3·2-s + 5·4-s + 5i·7-s + 3·8-s + 15i·11-s + 10i·13-s + 15i·14-s − 11·16-s + 18·17-s + 16·19-s + 45i·22-s + 12·23-s + 30i·26-s + 25i·28-s + 30i·29-s + ⋯
L(s)  = 1  + 1.5·2-s + 1.25·4-s + 0.714i·7-s + 0.375·8-s + 1.36i·11-s + 0.769i·13-s + 1.07i·14-s − 0.687·16-s + 1.05·17-s + 0.842·19-s + 2.04i·22-s + 0.521·23-s + 1.15i·26-s + 0.892i·28-s + 1.03i·29-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.447 - 0.894i$
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: \((2,\ 675,\ (\ :1),\ 0.447 - 0.894i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.943820892\)
\(L(\frac12)\) \(\approx\) \(3.943820892\)
\(L(2)\) 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 \( 1 - 3T + 4T^{2} \)
7 \( 1 - 5iT - 49T^{2} \)
11 \( 1 - 15iT - 121T^{2} \)
13 \( 1 - 10iT - 169T^{2} \)
17 \( 1 - 18T + 289T^{2} \)
19 \( 1 - 16T + 361T^{2} \)
23 \( 1 - 12T + 529T^{2} \)
29 \( 1 - 30iT - 841T^{2} \)
31 \( 1 + T + 961T^{2} \)
37 \( 1 - 20iT - 1.36e3T^{2} \)
41 \( 1 + 60iT - 1.68e3T^{2} \)
43 \( 1 + 50iT - 1.84e3T^{2} \)
47 \( 1 + 6T + 2.20e3T^{2} \)
53 \( 1 - 27T + 2.80e3T^{2} \)
59 \( 1 + 30iT - 3.48e3T^{2} \)
61 \( 1 + 76T + 3.72e3T^{2} \)
67 \( 1 + 10iT - 4.48e3T^{2} \)
71 \( 1 - 90iT - 5.04e3T^{2} \)
73 \( 1 + 65iT - 5.32e3T^{2} \)
79 \( 1 + 14T + 6.24e3T^{2} \)
83 \( 1 + 3T + 6.88e3T^{2} \)
89 \( 1 - 90iT - 7.92e3T^{2} \)
97 \( 1 + 85iT - 9.40e3T^{2} \)
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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

−10.64851599175318632055552668802, −9.582167197817375109862810087086, −8.842134294887871721329781376839, −7.40557003774436145508132022684, −6.76698480526142161823728617350, −5.57835892865768253019660573957, −5.04207731762770420139637553484, −4.01892325873428724013848435848, −2.98001924255302975424697300431, −1.84198216154308183204170517647, 0.894699824673261467153978948570, 2.92481409051240291336276720271, 3.49634623835504241710561409842, 4.58784017769784777864863934685, 5.59944052124229439036062918436, 6.14202149878752256502868416011, 7.34969937871699660532418298244, 8.155070555436612444889929464082, 9.388483494944059920257428679423, 10.42240909120571122998171852432

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