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Properties

Label 2-675-45.34-c1-0-13
Degree $2$
Conductor $675$
Sign $0.999 - 0.0234i$
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  + (1.41 + 0.816i)2-s + (0.334 + 0.579i)4-s + (−0.437 − 0.252i)7-s − 2.17i·8-s + (1.55 − 2.68i)11-s + (5.40 − 3.11i)13-s + (−0.412 − 0.714i)14-s + (2.44 − 4.23i)16-s + 6.10i·17-s + 5.57·19-s + (4.38 − 2.53i)22-s + (−3.31 + 1.91i)23-s + 10.1·26-s − 0.338i·28-s + (−1.22 + 2.12i)29-s + ⋯
L(s)  = 1  + (1.00 + 0.577i)2-s + (0.167 + 0.289i)4-s + (−0.165 − 0.0955i)7-s − 0.768i·8-s + (0.467 − 0.809i)11-s + (1.49 − 0.865i)13-s + (−0.110 − 0.191i)14-s + (0.611 − 1.05i)16-s + 1.47i·17-s + 1.27·19-s + (0.935 − 0.539i)22-s + (−0.690 + 0.398i)23-s + 1.99·26-s − 0.0638i·28-s + (−0.228 + 0.395i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.999 - 0.0234i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.999 - 0.0234i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55633 + 0.0299821i\)
\(L(\frac12)\) \(\approx\) \(2.55633 + 0.0299821i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good2 \( 1 + (-1.41 - 0.816i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (0.437 + 0.252i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.55 + 2.68i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.40 + 3.11i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 6.10iT - 17T^{2} \)
19 \( 1 - 5.57T + 19T^{2} \)
23 \( 1 + (3.31 - 1.91i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.22 - 2.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.11 + 3.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.72iT - 37T^{2} \)
41 \( 1 + (2.72 + 4.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.14 - 0.663i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (3.21 + 1.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 2.54iT - 53T^{2} \)
59 \( 1 + (1.44 + 2.49i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.42 + 2.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.08 + 1.20i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.54T + 71T^{2} \)
73 \( 1 - 11.7iT - 73T^{2} \)
79 \( 1 + (-1.70 + 2.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-12.0 - 6.95i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.38T + 89T^{2} \)
97 \( 1 + (9.59 + 5.53i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.55434292304799143889366975812, −9.693711776012385658300097903017, −8.591634851934832923311008311448, −7.80289265193539319058935531033, −6.56146026137070370986892389733, −5.95040629105731181215368333001, −5.27302051366957431618087500960, −3.77961227635432504253932656294, −3.46833610180295361130519208810, −1.20393051223804859859694704329, 1.66672873160173193136478011614, 2.98277858968512173707667446892, 3.93494198233096335588329285199, 4.77080500275513789704962718098, 5.77882580541541274718035714567, 6.79677177510048647019443623061, 7.82638842838172884133609181724, 8.972969361957093745044305184303, 9.601429687432143554240911581081, 10.83835125201534730069537687790

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