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Properties

Label 2-675-135.124-c1-0-43
Degree $2$
Conductor $675$
Sign $0.109 + 0.993i$
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  + (1.03 − 1.23i)2-s + (1.23 − 1.21i)3-s + (−0.105 − 0.597i)4-s + (−0.212 − 2.78i)6-s + (1.71 + 0.302i)7-s + (1.94 + 1.12i)8-s + (0.0676 − 2.99i)9-s + (2.93 − 1.06i)11-s + (−0.853 − 0.612i)12-s + (1.21 + 1.44i)13-s + (2.15 − 1.80i)14-s + (4.55 − 1.65i)16-s + (−5.56 + 3.21i)17-s + (−3.63 − 3.19i)18-s + (−2.43 + 4.21i)19-s + ⋯
L(s)  = 1  + (0.733 − 0.874i)2-s + (0.715 − 0.699i)3-s + (−0.0526 − 0.298i)4-s + (−0.0866 − 1.13i)6-s + (0.647 + 0.114i)7-s + (0.688 + 0.397i)8-s + (0.0225 − 0.999i)9-s + (0.884 − 0.321i)11-s + (−0.246 − 0.176i)12-s + (0.337 + 0.402i)13-s + (0.575 − 0.482i)14-s + (1.13 − 0.414i)16-s + (−1.34 + 0.778i)17-s + (−0.857 − 0.753i)18-s + (−0.558 + 0.967i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.109 + 0.993i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (124, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.29922 - 2.05963i\)
\(L(\frac12)\) \(\approx\) \(2.29922 - 2.05963i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.23 + 1.21i)T \)
5 \( 1 \)
good2 \( 1 + (-1.03 + 1.23i)T + (-0.347 - 1.96i)T^{2} \)
7 \( 1 + (-1.71 - 0.302i)T + (6.57 + 2.39i)T^{2} \)
11 \( 1 + (-2.93 + 1.06i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (-1.21 - 1.44i)T + (-2.25 + 12.8i)T^{2} \)
17 \( 1 + (5.56 - 3.21i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.43 - 4.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.73 - 0.481i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.94 + 4.98i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (0.687 + 3.89i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (0.704 - 0.406i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.80 + 4.86i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.901 + 2.47i)T + (-32.9 + 27.6i)T^{2} \)
47 \( 1 + (7.03 + 1.24i)T + (44.1 + 16.0i)T^{2} \)
53 \( 1 + 1.86iT - 53T^{2} \)
59 \( 1 + (0.971 + 0.353i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.78 - 10.1i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-8.76 - 10.4i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-4.06 - 7.03i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (8.53 + 4.92i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-11.3 - 9.56i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (5.92 - 7.06i)T + (-14.4 - 81.7i)T^{2} \)
89 \( 1 + (-6.28 + 10.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.39 - 14.8i)T + (-74.3 + 62.3i)T^{2} \)
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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

−10.59222256604962832598761198935, −9.381235023070166687969730680124, −8.460290181229207125376246120096, −7.88352272667074443441799877890, −6.70493648518675406553455365033, −5.77087312630096008690946615762, −4.12236282268814924399541828136, −3.83438007716546518105027965209, −2.27634193278807308696379066844, −1.64237742594467920450898533647, 1.87198864730275629675092479639, 3.45509220535545195524857332688, 4.57395793662979583730055541035, 4.89882560258140593896985780450, 6.24995230709484139308407568120, 7.09682729025215052426545002785, 7.980759936550222676419818468592, 8.939723581205023805014841435824, 9.636686647195102370198443578893, 10.84187428811258693405858759331

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