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Properties

Label 2-675-135.49-c1-0-22
Degree $2$
Conductor $675$
Sign $-0.362 + 0.932i$
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  + (−1.42 − 1.70i)2-s + (1.70 − 0.320i)3-s + (−0.511 + 2.89i)4-s + (−2.97 − 2.44i)6-s + (−3.70 + 0.653i)7-s + (1.81 − 1.04i)8-s + (2.79 − 1.09i)9-s + (5.03 + 1.83i)11-s + (0.0589 + 5.09i)12-s + (3.93 − 4.69i)13-s + (6.41 + 5.37i)14-s + (1.15 + 0.418i)16-s + (2.43 + 1.40i)17-s + (−5.85 − 3.20i)18-s + (−1.71 − 2.96i)19-s + ⋯
L(s)  = 1  + (−1.01 − 1.20i)2-s + (0.982 − 0.185i)3-s + (−0.255 + 1.44i)4-s + (−1.21 − 0.996i)6-s + (−1.40 + 0.247i)7-s + (0.641 − 0.370i)8-s + (0.931 − 0.363i)9-s + (1.51 + 0.553i)11-s + (0.0170 + 1.47i)12-s + (1.09 − 1.30i)13-s + (1.71 + 1.43i)14-s + (0.287 + 0.104i)16-s + (0.589 + 0.340i)17-s + (−1.37 − 0.754i)18-s + (−0.393 − 0.681i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $-0.362 + 0.932i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ -0.362 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.665337 - 0.972518i\)
\(L(\frac12)\) \(\approx\) \(0.665337 - 0.972518i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

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

−10.08181212510475441724413584082, −9.325828788850316624717571347205, −8.911656017656748780097829467105, −8.045872589030457850162202489447, −6.93217184601354147421938843777, −5.97911040147372051422997110987, −3.80599388524611237903436889458, −3.37751691766840632194666333598, −2.24148526823353617726522244514, −0.939920358608960528037839119259, 1.33066123771066717027847336809, 3.35313602038144791097321854119, 4.05430014462604682000615639103, 5.95528391893248219271491349198, 6.62553526316725775096181293753, 7.18137378494448587825438748223, 8.459113059813784478797327271551, 8.835148403236242816895494942125, 9.644994564916589050202389340141, 10.05337736569700068429788285373

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