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Properties

Label 2-675-45.34-c1-0-3
Degree $2$
Conductor $675$
Sign $0.421 - 0.907i$
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  + (−1.27 − 0.736i)2-s + (0.0852 + 0.147i)4-s + (3.34 + 1.93i)7-s + 2.69i·8-s + (0.130 − 0.225i)11-s + (−3.53 + 2.03i)13-s + (−2.84 − 4.93i)14-s + (2.15 − 3.73i)16-s + 3.26i·17-s − 4.24·19-s + (−0.332 + 0.191i)22-s + (−7.53 + 4.34i)23-s + 6.00·26-s + 0.659i·28-s + (−2.11 + 3.65i)29-s + ⋯
L(s)  = 1  + (−0.902 − 0.520i)2-s + (0.0426 + 0.0738i)4-s + (1.26 + 0.730i)7-s + 0.952i·8-s + (0.0392 − 0.0679i)11-s + (−0.979 + 0.565i)13-s + (−0.761 − 1.31i)14-s + (0.538 − 0.933i)16-s + 0.790i·17-s − 0.974·19-s + (−0.0708 + 0.0408i)22-s + (−1.57 + 0.906i)23-s + 1.17·26-s + 0.124i·28-s + (−0.392 + 0.678i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.421 - 0.907i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.421 - 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.421 - 0.907i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (424, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.421 - 0.907i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.557857 + 0.356068i\)
\(L(\frac12)\) \(\approx\) \(0.557857 + 0.356068i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 0.736i)T + (1 + 1.73i)T^{2} \)
7 \( 1 + (-3.34 - 1.93i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.130 + 0.225i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.53 - 2.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.26iT - 17T^{2} \)
19 \( 1 + 4.24T + 19T^{2} \)
23 \( 1 + (7.53 - 4.34i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.11 - 3.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.32 + 2.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2.27iT - 37T^{2} \)
41 \( 1 + (-2.82 - 4.88i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-7.85 - 4.53i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.23 - 0.714i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.3iT - 53T^{2} \)
59 \( 1 + (-3.56 - 6.16i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.26 - 2.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.77 - 5.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.38T + 71T^{2} \)
73 \( 1 - 0.403iT - 73T^{2} \)
79 \( 1 + (1.52 - 2.63i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.96 - 2.29i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 7.17T + 89T^{2} \)
97 \( 1 + (-2.69 - 1.55i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.59627928556882003630880647367, −9.769081284608532224293759918696, −8.984030810260001684186217444612, −8.235417129529447381715901595012, −7.60019831911989074199485277037, −6.08622121244743590449428695242, −5.20507490471642916766574340672, −4.21305847576428769304697801518, −2.34502382583264970406955766396, −1.64606738584323353724614590057, 0.47370394659779834744148057897, 2.19595496431229330938033752311, 3.97201971735377572445890915080, 4.75111256770640815192599625289, 6.06857640335613893471335324849, 7.29487399729024841031203280702, 7.68787086850938979120467415584, 8.461928599109841709095116331495, 9.352504747516042123288516210261, 10.29946912807214627252772281045

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