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Properties

Label 2-675-135.79-c1-0-18
Degree $2$
Conductor $675$
Sign $0.866 + 0.498i$
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  + (−1.32 − 0.233i)2-s + (0.300 + 1.70i)3-s + (−0.173 − 0.0632i)4-s − 2.33i·6-s + (−0.237 − 0.652i)7-s + (2.54 + 1.47i)8-s + (−2.81 + 1.02i)9-s + (−3.52 − 2.95i)11-s + (0.0555 − 0.315i)12-s + (1.39 − 0.245i)13-s + (0.162 + 0.921i)14-s + (−2.75 − 2.31i)16-s + (3.35 − 1.93i)17-s + (3.98 − 0.701i)18-s + (3.53 − 6.11i)19-s + ⋯
L(s)  = 1  + (−0.938 − 0.165i)2-s + (0.173 + 0.984i)3-s + (−0.0868 − 0.0316i)4-s − 0.952i·6-s + (−0.0897 − 0.246i)7-s + (0.901 + 0.520i)8-s + (−0.939 + 0.342i)9-s + (−1.06 − 0.890i)11-s + (0.0160 − 0.0909i)12-s + (0.385 − 0.0679i)13-s + (0.0434 + 0.246i)14-s + (−0.688 − 0.577i)16-s + (0.814 − 0.470i)17-s + (0.938 − 0.165i)18-s + (0.810 − 1.40i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.866 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.866 + 0.498i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.866 + 0.498i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.695868 - 0.185796i\)
\(L(\frac12)\) \(\approx\) \(0.695868 - 0.185796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.300 - 1.70i)T \)
5 \( 1 \)
good2 \( 1 + (1.32 + 0.233i)T + (1.87 + 0.684i)T^{2} \)
7 \( 1 + (0.237 + 0.652i)T + (-5.36 + 4.49i)T^{2} \)
11 \( 1 + (3.52 + 2.95i)T + (1.91 + 10.8i)T^{2} \)
13 \( 1 + (-1.39 + 0.245i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-3.35 + 1.93i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.53 + 6.11i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.30 - 3.59i)T + (-17.6 - 14.7i)T^{2} \)
29 \( 1 + (0.851 - 4.82i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-0.786 - 0.286i)T + (23.7 + 19.9i)T^{2} \)
37 \( 1 + (-6.91 + 3.99i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.36 + 7.74i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (-1.33 + 1.59i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (-2.35 - 6.46i)T + (-36.0 + 30.2i)T^{2} \)
53 \( 1 - 3.05iT - 53T^{2} \)
59 \( 1 + (-6.82 + 5.72i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (-8.12 + 2.95i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-9.30 + 1.64i)T + (62.9 - 22.9i)T^{2} \)
71 \( 1 + (-2.90 - 5.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.68 + 2.70i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.27 + 12.9i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (1.11 + 0.197i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (-0.368 + 0.637i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.36 - 6.39i)T + (-16.8 - 95.5i)T^{2} \)
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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

−10.32528780555011337614720903119, −9.522426375888190694028423593847, −8.928661338589449477849925071449, −8.086160729585693659491367173085, −7.31869722782587151999877634616, −5.58184593789101889754122754338, −5.06748380628653155979175666852, −3.74744256377714989157463753481, −2.66085796499277951586985337821, −0.62849322601942629842673597837, 1.14658906272257958365919149968, 2.45189780804244914999831040983, 3.91980607458642315216404537444, 5.35730637163232270355990421131, 6.35241060450892734967647982055, 7.46975732032247648366759491313, 7.978210143242003999672557735940, 8.547316996741246588766523143508, 9.807624564525901254251490813176, 10.12444590071057972626949905274

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