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Properties

Label 2-675-27.22-c1-0-42
Degree $2$
Conductor $675$
Sign $-0.835 + 0.549i$
Analytic cond. $5.38990$
Root an. cond. $2.32161$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.93 + 1.62i)2-s + (1.11 − 1.32i)3-s + (0.766 − 4.34i)4-s + 4.38i·6-s + (0.532 + 3.01i)7-s + (3.05 + 5.28i)8-s + (−0.520 − 2.95i)9-s + (−5.29 − 1.92i)11-s + (−4.91 − 5.85i)12-s + (−3.23 − 2.71i)13-s + (−5.94 − 4.98i)14-s + (−6.23 − 2.27i)16-s + (−0.826 + 1.43i)17-s + (5.81 + 4.88i)18-s + (−0.120 − 0.208i)19-s + ⋯
L(s)  = 1  + (−1.37 + 1.15i)2-s + (0.642 − 0.766i)3-s + (0.383 − 2.17i)4-s + 1.79i·6-s + (0.201 + 1.14i)7-s + (1.07 + 1.86i)8-s + (−0.173 − 0.984i)9-s + (−1.59 − 0.581i)11-s + (−1.41 − 1.68i)12-s + (−0.898 − 0.753i)13-s + (−1.58 − 1.33i)14-s + (−1.55 − 0.567i)16-s + (−0.200 + 0.347i)17-s + (1.37 + 1.15i)18-s + (−0.0276 − 0.0479i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.835 + 0.549i$
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: \(1\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.835 + 0.549i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.11 + 1.32i)T \)
5 \( 1 \)
good2 \( 1 + (1.93 - 1.62i)T + (0.347 - 1.96i)T^{2} \)
7 \( 1 + (-0.532 - 3.01i)T + (-6.57 + 2.39i)T^{2} \)
11 \( 1 + (5.29 + 1.92i)T + (8.42 + 7.07i)T^{2} \)
13 \( 1 + (3.23 + 2.71i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (0.826 - 1.43i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.120 + 0.208i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.29 - 7.34i)T + (-21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.90 - 4.95i)T + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.858 + 4.86i)T + (-29.1 - 10.6i)T^{2} \)
37 \( 1 + (1.24 - 2.15i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.109 + 0.0918i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (0.705 + 0.256i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (0.807 + 4.58i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + 12.1T + 53T^{2} \)
59 \( 1 + (-4.45 + 1.62i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-2.41 - 13.6i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (5.64 + 4.73i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-2.45 + 4.24i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.113 + 0.196i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (7.53 - 6.32i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-6.78 + 5.69i)T + (14.4 - 81.7i)T^{2} \)
89 \( 1 + (3.33 + 5.76i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (8.95 + 3.26i)T + (74.3 + 62.3i)T^{2} \)
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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

−9.706839310991265468453715504199, −9.039221418787992652726388355748, −8.139671094459287681237948961610, −7.84001153927305525207352732302, −6.99479171343914186905917411908, −5.72097676974394220050991152078, −5.43214694612245538633327461469, −2.96594940977840210745404229508, −1.84022528968094277789248127366, 0, 2.03496218524283038588272824679, 2.81845613612345684662714316181, 4.09442998754590062730488525548, 4.89984233320482196645716659628, 7.09259488986514202234756640025, 7.77370930172698959170044551603, 8.396772974160628898683157438704, 9.520238079609604258959980100337, 9.919870290748362556322280055151

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