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Properties

Label 2-675-15.14-c2-0-45
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  + 3.31·2-s + 7·4-s − 9i·7-s + 9.94·8-s − 13.2i·11-s − 10i·13-s − 29.8i·14-s + 5.00·16-s − 26.5·17-s + 29·19-s − 44i·22-s + 39.7·23-s − 33.1i·26-s − 63i·28-s + 26.5i·29-s + ⋯
L(s)  = 1  + 1.65·2-s + 1.75·4-s − 1.28i·7-s + 1.24·8-s − 1.20i·11-s − 0.769i·13-s − 2.13i·14-s + 0.312·16-s − 1.56·17-s + 1.52·19-s − 2i·22-s + 1.73·23-s − 1.27i·26-s − 2.25i·28-s + 0.914i·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

\(L(\frac{3}{2})\) \(\approx\) \(4.572187657\)
\(L(\frac12)\) \(\approx\) \(4.572187657\)
\(L(2)\) 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 \( 1 - 3.31T + 4T^{2} \)
7 \( 1 + 9iT - 49T^{2} \)
11 \( 1 + 13.2iT - 121T^{2} \)
13 \( 1 + 10iT - 169T^{2} \)
17 \( 1 + 26.5T + 289T^{2} \)
19 \( 1 - 29T + 361T^{2} \)
23 \( 1 - 39.7T + 529T^{2} \)
29 \( 1 - 26.5iT - 841T^{2} \)
31 \( 1 - 15T + 961T^{2} \)
37 \( 1 - 59iT - 1.36e3T^{2} \)
41 \( 1 - 13.2iT - 1.68e3T^{2} \)
43 \( 1 - 5iT - 1.84e3T^{2} \)
47 \( 1 - 13.2T + 2.20e3T^{2} \)
53 \( 1 + 13.2T + 2.80e3T^{2} \)
59 \( 1 + 66.3iT - 3.48e3T^{2} \)
61 \( 1 - 19T + 3.72e3T^{2} \)
67 \( 1 + 54iT - 4.48e3T^{2} \)
71 \( 1 - 92.8iT - 5.04e3T^{2} \)
73 \( 1 + 55iT - 5.32e3T^{2} \)
79 \( 1 - 33T + 6.24e3T^{2} \)
83 \( 1 - 79.5T + 6.88e3T^{2} \)
89 \( 1 - 119. iT - 7.92e3T^{2} \)
97 \( 1 + 97iT - 9.40e3T^{2} \)
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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

−10.71479806286390526276297936777, −9.352486816826627308639377889272, −8.176396782914396242002835361116, −7.06329898957489800268875280427, −6.50157394032459423973290137834, −5.32307137012240494034608627037, −4.66985418209471478824819346401, −3.51219660737484474604789266458, −2.93726541814227796793517080904, −0.974841992387178717693703864387, 2.05558665121985302428327278697, 2.79483436767701488264957715509, 4.15205092312095349499927769405, 4.92357157446188975655344286983, 5.67790025426564816432861833413, 6.69357035780664464509291616559, 7.34609799279355297378584268473, 8.929372433872187479015312017002, 9.434487169172113888128121126878, 10.89112257742651715196453273848

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