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Properties

Label 2-675-25.11-c1-0-18
Degree $2$
Conductor $675$
Sign $0.866 - 0.499i$
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  + (−0.634 + 0.460i)2-s + (−0.428 + 1.31i)4-s + (−2.22 − 0.258i)5-s + 4.26·7-s + (−0.820 − 2.52i)8-s + (1.52 − 0.859i)10-s + (3.00 − 2.17i)11-s + (−1.18 − 0.859i)13-s + (−2.70 + 1.96i)14-s + (−0.558 − 0.405i)16-s + (−0.0757 − 0.232i)17-s + (−0.130 − 0.401i)19-s + (1.29 − 2.81i)20-s + (−0.898 + 2.76i)22-s + (3.66 − 2.66i)23-s + ⋯
L(s)  = 1  + (−0.448 + 0.325i)2-s + (−0.214 + 0.658i)4-s + (−0.993 − 0.115i)5-s + 1.61·7-s + (−0.289 − 0.892i)8-s + (0.483 − 0.271i)10-s + (0.904 − 0.657i)11-s + (−0.328 − 0.238i)13-s + (−0.723 + 0.525i)14-s + (−0.139 − 0.101i)16-s + (−0.0183 − 0.0565i)17-s + (−0.0299 − 0.0920i)19-s + (0.288 − 0.629i)20-s + (−0.191 + 0.589i)22-s + (0.764 − 0.555i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.866 - 0.499i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (136, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.866 - 0.499i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09396 + 0.292548i\)
\(L(\frac12)\) \(\approx\) \(1.09396 + 0.292548i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (2.22 + 0.258i)T \)
good2 \( 1 + (0.634 - 0.460i)T + (0.618 - 1.90i)T^{2} \)
7 \( 1 - 4.26T + 7T^{2} \)
11 \( 1 + (-3.00 + 2.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (1.18 + 0.859i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.0757 + 0.232i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (0.130 + 0.401i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-3.66 + 2.66i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (1.55 - 4.79i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (0.576 + 1.77i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.03 - 4.38i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-8.72 - 6.33i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + (0.00130 - 0.00401i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.61 - 8.06i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (9.91 + 7.20i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (1.66 - 1.21i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.89 + 5.84i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-2.78 + 8.57i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (7.26 - 5.27i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-5.18 + 15.9i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-4.72 - 14.5i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.244 - 0.177i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.42 + 7.47i)T + (-78.4 - 57.0i)T^{2} \)
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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

−10.88391152448532061760804076051, −9.258334507798434203860970728315, −8.751636559554171294436127235036, −7.82379270663102713790501566072, −7.55509808207721473325337033956, −6.32505797875105867543364634439, −4.81432617132585997518711912473, −4.19374996337249798983606864642, −3.00288106278513476861003163674, −1.00075896155255855688846830287, 1.08180491867497092809406223567, 2.25799042869068784868147796056, 4.12172401904608795114173158169, 4.72734747160683020696144345291, 5.81696230178916949123996619430, 7.21808392349796055437564280586, 7.85343034881389681704413481952, 8.836772457135889956652637817148, 9.446986614456742080401342424038, 10.63388776994399856143938276951

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