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Properties

Label 2-675-27.16-c1-0-8
Degree $2$
Conductor $675$
Sign $0.992 - 0.124i$
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  + (−1.01 − 0.852i)2-s + (0.196 − 1.72i)3-s + (−0.0418 − 0.237i)4-s + (−1.66 + 1.58i)6-s + (−0.769 + 4.36i)7-s + (−1.48 + 2.57i)8-s + (−2.92 − 0.674i)9-s + (3.91 − 1.42i)11-s + (−0.416 + 0.0254i)12-s + (−1.15 + 0.972i)13-s + (4.50 − 3.77i)14-s + (3.25 − 1.18i)16-s + (0.568 + 0.985i)17-s + (2.39 + 3.17i)18-s + (−3.37 + 5.84i)19-s + ⋯
L(s)  = 1  + (−0.718 − 0.602i)2-s + (0.113 − 0.993i)3-s + (−0.0209 − 0.118i)4-s + (−0.680 + 0.645i)6-s + (−0.291 + 1.65i)7-s + (−0.525 + 0.910i)8-s + (−0.974 − 0.224i)9-s + (1.17 − 0.429i)11-s + (−0.120 + 0.00735i)12-s + (−0.321 + 0.269i)13-s + (1.20 − 1.01i)14-s + (0.812 − 0.295i)16-s + (0.137 + 0.238i)17-s + (0.564 + 0.748i)18-s + (−0.774 + 1.34i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 - 0.124i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.992 - 0.124i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.992 - 0.124i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.727102 + 0.0455687i\)
\(L(\frac12)\) \(\approx\) \(0.727102 + 0.0455687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.196 + 1.72i)T \)
5 \( 1 \)
good2 \( 1 + (1.01 + 0.852i)T + (0.347 + 1.96i)T^{2} \)
7 \( 1 + (0.769 - 4.36i)T + (-6.57 - 2.39i)T^{2} \)
11 \( 1 + (-3.91 + 1.42i)T + (8.42 - 7.07i)T^{2} \)
13 \( 1 + (1.15 - 0.972i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.568 - 0.985i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.37 - 5.84i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.38 - 7.86i)T + (-21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.89 - 1.58i)T + (5.03 + 28.5i)T^{2} \)
31 \( 1 + (-0.369 - 2.09i)T + (-29.1 + 10.6i)T^{2} \)
37 \( 1 + (4.99 + 8.65i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.384 - 0.322i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (0.102 - 0.0373i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.31 + 7.45i)T + (-44.1 - 16.0i)T^{2} \)
53 \( 1 - 3.96T + 53T^{2} \)
59 \( 1 + (-7.67 - 2.79i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (2.14 - 12.1i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.25 - 1.88i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (3.78 + 6.56i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.12 - 8.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.74 - 3.14i)T + (13.7 + 77.7i)T^{2} \)
83 \( 1 + (2.78 + 2.34i)T + (14.4 + 81.7i)T^{2} \)
89 \( 1 + (-5.74 + 9.95i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (7.36 - 2.68i)T + (74.3 - 62.3i)T^{2} \)
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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

−10.48921700559530753948990031020, −9.398769314908662327968015619874, −8.857952632272465125490908783422, −8.327005366724167791213020037536, −7.02133684116257462502388942758, −5.89872442982806987612776979573, −5.57816963142236041174722253522, −3.48331344971189240677768288235, −2.27088256121941055234924192139, −1.45694438158152312893269646198, 0.50736953889802604768916836579, 3.01651975608031461906295509900, 4.11095164313952712896450314384, 4.65043299326080376589851286177, 6.50261907432109747227787523331, 6.89863012216865603043214078898, 7.987303993594420521245335860476, 8.816971863171235374917458027389, 9.579006712196541704690545239189, 10.22742481391169073451398159729

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