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Properties

Label 2-675-27.22-c1-0-52
Degree $2$
Conductor $675$
Sign $-0.995 + 0.0927i$
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  + (1.70 − 1.42i)2-s + (−0.320 − 1.70i)3-s + (0.511 − 2.89i)4-s + (−2.97 − 2.44i)6-s + (−0.653 − 3.70i)7-s + (−1.04 − 1.81i)8-s + (−2.79 + 1.09i)9-s + (5.03 + 1.83i)11-s + (−5.09 + 0.0589i)12-s + (−4.69 − 3.93i)13-s + (−6.41 − 5.37i)14-s + (1.15 + 0.418i)16-s + (−1.40 + 2.43i)17-s + (−3.20 + 5.85i)18-s + (1.71 + 2.96i)19-s + ⋯
L(s)  = 1  + (1.20 − 1.01i)2-s + (−0.185 − 0.982i)3-s + (0.255 − 1.44i)4-s + (−1.21 − 0.996i)6-s + (−0.247 − 1.40i)7-s + (−0.370 − 0.641i)8-s + (−0.931 + 0.363i)9-s + (1.51 + 0.553i)11-s + (−1.47 + 0.0170i)12-s + (−1.30 − 1.09i)13-s + (−1.71 − 1.43i)14-s + (0.287 + 0.104i)16-s + (−0.340 + 0.589i)17-s + (−0.754 + 1.37i)18-s + (0.393 + 0.681i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.995 + 0.0927i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.995 + 0.0927i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.116680 - 2.51140i\)
\(L(\frac12)\) \(\approx\) \(0.116680 - 2.51140i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.320 + 1.70i)T \)
5 \( 1 \)
good2 \( 1 + (-1.70 + 1.42i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (0.653 + 3.70i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (-5.03 - 1.83i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (4.69 + 3.93i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.40 - 2.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.71 - 2.96i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.155 - 0.882i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (-1.61 + 1.35i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.576 + 3.26i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (-1.73 + 3.00i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.85 + 1.55i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (6.42 + 2.33i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-0.626 - 3.55i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 + (5.74 - 2.09i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-0.0966 - 0.548i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-12.1 - 10.1i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-7.71 + 13.3i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.47 - 11.2i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.60 + 6.38i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3.66 + 3.07i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (1.45 + 2.51i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.99 + 2.18i)T + (74.3 + 62.3i)T^{2} \)
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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

−10.37055184539396328520706756977, −9.661805692038784100276846032744, −8.071155303012069804588355302635, −7.24375790701141922350258555158, −6.41673511614254238735354432318, −5.36115020465969817642432266686, −4.26801388421407642140901737897, −3.45527524867496362371079869162, −2.15608076410624486826964841180, −0.986997655536987028811533819836, 2.68939643962615577487686389785, 3.74987916670377856179644142567, 4.79800611786821973902084651512, 5.28117760173947119133811424404, 6.44197961181818771252231759801, 6.78820922805597929433731328797, 8.380635737934760442420791639369, 9.234848912119284310574799006068, 9.696918530828936192685359830616, 11.30040238170812720096486583454

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