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Properties

Label 2-675-9.7-c1-0-9
Degree $2$
Conductor $675$
Sign $0.595 + 0.803i$
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  + (−1.25 + 2.17i)2-s + (−2.16 − 3.74i)4-s + (−0.257 + 0.445i)7-s + 5.83·8-s + (−1.66 + 2.87i)11-s + (−0.660 − 1.14i)13-s + (−0.646 − 1.11i)14-s + (−3.01 + 5.22i)16-s − 3.32·17-s − 1.32·19-s + (−4.17 − 7.23i)22-s + (−2.06 − 3.57i)23-s + 3.32·26-s + 2.22·28-s + (−0.693 + 1.20i)29-s + ⋯
L(s)  = 1  + (−0.888 + 1.53i)2-s + (−1.08 − 1.87i)4-s + (−0.0971 + 0.168i)7-s + 2.06·8-s + (−0.500 + 0.867i)11-s + (−0.183 − 0.317i)13-s + (−0.172 − 0.299i)14-s + (−0.753 + 1.30i)16-s − 0.805·17-s − 0.303·19-s + (−0.890 − 1.54i)22-s + (−0.430 − 0.745i)23-s + 0.651·26-s + 0.419·28-s + (−0.128 + 0.222i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.595 + 0.803i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.595 + 0.803i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.595 + 0.803i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.151302 - 0.0762082i\)
\(L(\frac12)\) \(\approx\) \(0.151302 - 0.0762082i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.25 - 2.17i)T + (-1 - 1.73i)T^{2} \)
7 \( 1 + (0.257 - 0.445i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.66 - 2.87i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.660 + 1.14i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 3.32T + 17T^{2} \)
19 \( 1 + 1.32T + 19T^{2} \)
23 \( 1 + (2.06 + 3.57i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.693 - 1.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (4.36 + 7.56i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.292T + 37T^{2} \)
41 \( 1 + (5.67 + 9.82i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.17 + 8.96i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.43 + 4.21i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 5.02T + 53T^{2} \)
59 \( 1 + (2.51 + 4.35i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.67 - 6.36i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.72 - 8.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 8.99T + 71T^{2} \)
73 \( 1 + 6.05T + 73T^{2} \)
79 \( 1 + (-4.02 + 6.97i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.771 - 1.33i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (6.12 - 10.6i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.13814898538878533740929239386, −9.190951332440742806523464121799, −8.604511452173781805202304358472, −7.59920587065523431602482155485, −7.07339385492898213723954687590, −6.07516870060887400959711536572, −5.27707201120011134649945293660, −4.22945125352899184850075259714, −2.19740894355872632037123980973, −0.12071564774306242017003622109, 1.50283543159909682813111091049, 2.73715037349610513840168530819, 3.64500975778967013080768340106, 4.78177851836079994314696131305, 6.23518883236906577892360513863, 7.53903692349321517755555575038, 8.376433017271665840814665942565, 9.111864215430295811881893306792, 9.858532044225941022271358248443, 10.74651688250928569446287698472

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