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Properties

Label 2-675-9.2-c2-0-32
Degree $2$
Conductor $675$
Sign $-0.999 + 0.00852i$
Analytic cond. $18.3924$
Root an. cond. $4.28863$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.457 + 0.264i)2-s + (−1.86 − 3.22i)4-s + (−1.38 + 2.39i)7-s − 4.08i·8-s + (7.99 + 4.61i)11-s + (−6.79 − 11.7i)13-s + (−1.26 + 0.731i)14-s + (−6.36 + 11.0i)16-s − 12.2i·17-s − 20.2·19-s + (2.43 + 4.22i)22-s + (−2.05 + 1.18i)23-s − 7.18i·26-s + 10.2·28-s + (30.2 + 17.4i)29-s + ⋯
L(s)  = 1  + (0.228 + 0.132i)2-s + (−0.465 − 0.805i)4-s + (−0.197 + 0.342i)7-s − 0.510i·8-s + (0.726 + 0.419i)11-s + (−0.522 − 0.905i)13-s + (−0.0904 + 0.0522i)14-s + (−0.397 + 0.688i)16-s − 0.718i·17-s − 1.06·19-s + (0.110 + 0.192i)22-s + (−0.0892 + 0.0515i)23-s − 0.276i·26-s + 0.367·28-s + (1.04 + 0.601i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00852i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.999 + 0.00852i$
Analytic conductor: \(18.3924\)
Root analytic conductor: \(4.28863\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1),\ -0.999 + 0.00852i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3460774049\)
\(L(\frac12)\) \(\approx\) \(0.3460774049\)
\(L(2)\) 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 \( 1 + (-0.457 - 0.264i)T + (2 + 3.46i)T^{2} \)
7 \( 1 + (1.38 - 2.39i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (-7.99 - 4.61i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + (6.79 + 11.7i)T + (-84.5 + 146. i)T^{2} \)
17 \( 1 + 12.2iT - 289T^{2} \)
19 \( 1 + 20.2T + 361T^{2} \)
23 \( 1 + (2.05 - 1.18i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (-30.2 - 17.4i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (14.7 + 25.5i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 + 64.3T + 1.36e3T^{2} \)
41 \( 1 + (34.5 - 19.9i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (33.7 - 58.5i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (80.8 + 46.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 + 9.82iT - 2.80e3T^{2} \)
59 \( 1 + (50.6 - 29.2i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-7.75 + 13.4i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-7.78 - 13.4i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 53.1iT - 5.04e3T^{2} \)
73 \( 1 + 23.6T + 5.32e3T^{2} \)
79 \( 1 + (-17.2 + 29.9i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (65.2 + 37.6i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 + 29.1iT - 7.92e3T^{2} \)
97 \( 1 + (31.1 - 54.0i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−9.898188317464680805056081922317, −9.109518722351270878201401847367, −8.256911217951841876762208409698, −6.97006849944573660692113008289, −6.25858557128659532460998169184, −5.21344242487303332622850571942, −4.51244967766986156311414357959, −3.19370028467557535174468491081, −1.68983435835903265665761853094, −0.11224429400179295998414035093, 1.90933799939115721989664211274, 3.36901458168710350375858477466, 4.11613755794203260715131891822, 5.04089682468210736878829839143, 6.45588348959467466110353162174, 7.10728251018830891647240951566, 8.402690216500962605022195301525, 8.760345849378072802284287415407, 9.857668171968093388147146067228, 10.75477465518636463443722433453

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