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Properties

Label 2-675-135.4-c1-0-46
Degree $2$
Conductor $675$
Sign $0.117 + 0.993i$
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  + (0.300 − 0.826i)2-s + (1.62 − 0.592i)3-s + (0.939 + 0.788i)4-s − 1.52i·6-s + (−2.41 − 2.87i)7-s + (2.45 − 1.41i)8-s + (2.29 − 1.92i)9-s + (−0.180 − 1.02i)11-s + (1.99 + 0.726i)12-s + (−1.08 − 2.99i)13-s + (−3.10 + 1.13i)14-s + (−0.00727 − 0.0412i)16-s + (−0.405 − 0.233i)17-s + (−0.902 − 2.47i)18-s + (2.34 + 4.06i)19-s + ⋯
L(s)  = 1  + (0.212 − 0.584i)2-s + (0.939 − 0.342i)3-s + (0.469 + 0.394i)4-s − 0.621i·6-s + (−0.913 − 1.08i)7-s + (0.868 − 0.501i)8-s + (0.766 − 0.642i)9-s + (−0.0545 − 0.309i)11-s + (0.576 + 0.209i)12-s + (−0.302 − 0.830i)13-s + (−0.830 + 0.302i)14-s + (−0.00181 − 0.0103i)16-s + (−0.0982 − 0.0567i)17-s + (−0.212 − 0.584i)18-s + (0.538 + 0.932i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.117 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.117 + 0.993i$
Analytic conductor: \(5.38990\)
Root analytic conductor: \(2.32161\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (274, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :1/2),\ 0.117 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84900 - 1.64254i\)
\(L(\frac12)\) \(\approx\) \(1.84900 - 1.64254i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.62 + 0.592i)T \)
5 \( 1 \)
good2 \( 1 + (-0.300 + 0.826i)T + (-1.53 - 1.28i)T^{2} \)
7 \( 1 + (2.41 + 2.87i)T + (-1.21 + 6.89i)T^{2} \)
11 \( 1 + (0.180 + 1.02i)T + (-10.3 + 3.76i)T^{2} \)
13 \( 1 + (1.08 + 2.99i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (0.405 + 0.233i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.34 - 4.06i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.45 - 4.11i)T + (-3.99 - 22.6i)T^{2} \)
29 \( 1 + (-5.45 - 1.98i)T + (22.2 + 18.6i)T^{2} \)
31 \( 1 + (3.14 + 2.63i)T + (5.38 + 30.5i)T^{2} \)
37 \( 1 + (-3.87 - 2.23i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (7.52 - 2.73i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-11.9 + 2.11i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.22 + 2.65i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 - 8.83iT - 53T^{2} \)
59 \( 1 + (2.36 - 13.4i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (-7.46 + 6.26i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (0.623 + 1.71i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (3.85 - 6.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-0.705 + 0.407i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (3.81 + 1.38i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (5.81 - 15.9i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (5.19 + 9.00i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.02 - 1.06i)T + (91.1 - 33.1i)T^{2} \)
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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

−10.11072758669553975315984201190, −9.816878904203814502196050393020, −8.417366991611189809231459640121, −7.57071715066642209724476291327, −7.09354123535797082659809558875, −5.96869599337755112343967603580, −4.17651916823089099576722728218, −3.45827122602436038115954886676, −2.70252703236316582122298059914, −1.21192609207470900952189099355, 2.08399801724437977880282913809, 2.85449468057580580454719898934, 4.33565536775290560216198864350, 5.28307184201176531075440757803, 6.40126106107705162790474540269, 7.03917308345942727636341597833, 8.086856139536791214641369968187, 9.037621464340035479490657395462, 9.660628387202151001919311942489, 10.46600348238192142302371930962

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