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Properties

Label 2-675-27.16-c1-0-4
Degree $2$
Conductor $675$
Sign $0.998 - 0.0498i$
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  + (−2.06 − 1.73i)2-s + (0.239 − 1.71i)3-s + (0.918 + 5.20i)4-s + (−3.47 + 3.13i)6-s + (−0.175 + 0.996i)7-s + (4.43 − 7.68i)8-s + (−2.88 − 0.820i)9-s + (−1.18 + 0.430i)11-s + (9.15 − 0.329i)12-s + (−3.17 + 2.66i)13-s + (2.09 − 1.75i)14-s + (−12.5 + 4.57i)16-s + (−1.95 − 3.38i)17-s + (4.54 + 6.70i)18-s + (0.433 − 0.750i)19-s + ⋯
L(s)  = 1  + (−1.46 − 1.22i)2-s + (0.138 − 0.990i)3-s + (0.459 + 2.60i)4-s + (−1.41 + 1.27i)6-s + (−0.0664 + 0.376i)7-s + (1.56 − 2.71i)8-s + (−0.961 − 0.273i)9-s + (−0.356 + 0.129i)11-s + (2.64 − 0.0952i)12-s + (−0.880 + 0.739i)13-s + (0.559 − 0.469i)14-s + (−3.14 + 1.14i)16-s + (−0.474 − 0.821i)17-s + (1.07 + 1.58i)18-s + (0.0993 − 0.172i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.998 - 0.0498i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.998 - 0.0498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.368350 + 0.00919563i\)
\(L(\frac12)\) \(\approx\) \(0.368350 + 0.00919563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.239 + 1.71i)T \)
5 \( 1 \)
good2 \( 1 + (2.06 + 1.73i)T + (0.347 + 1.96i)T^{2} \)
7 \( 1 + (0.175 - 0.996i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (1.18 - 0.430i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (3.17 - 2.66i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (1.95 + 3.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.433 + 0.750i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.479 - 2.71i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-6.00 - 5.03i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.941 - 5.33i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (-3.71 - 6.44i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.73 - 2.29i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (4.00 - 1.45i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (0.640 - 3.63i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 0.485T + 53T^{2} \)
59 \( 1 + (-1.47 - 0.538i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (-0.286 + 1.62i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-2.53 + 2.13i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (4.95 + 8.58i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (2.63 - 4.55i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.41 + 5.38i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (-2.21 - 1.86i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (8.03 - 13.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.90 - 1.05i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.45613084547006875915563796485, −9.538826123883989794354210057534, −8.910527035391282083615866360824, −8.144952095988018313366750412236, −7.27140389974357544329965241114, −6.65102844666516911178581314378, −4.83588270903993844938924320554, −3.11087147476655214349235171464, −2.43511901441789221221902415681, −1.28892364398724743873092196962, 0.32783079882755944033679010685, 2.44324025691426192861637850401, 4.29004792827401180241815629284, 5.36249353829220273861680560211, 6.11946600105320931461711854403, 7.21477718374793172774331505496, 8.105106367574262627473061740368, 8.620939554280528928116063416757, 9.621697524038911002325025713101, 10.24169001091875558164164541425

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