Properties

Label 2-4050-5.4-c1-0-40
Degree $2$
Conductor $4050$
Sign $0.894 - 0.447i$
Analytic cond. $32.3394$
Root an. cond. $5.68677$
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  + i·2-s − 4-s − 2i·7-s i·8-s + 3·11-s + 2i·13-s + 2·14-s + 16-s − 3i·17-s + 19-s + 3i·22-s + 6i·23-s − 2·26-s + 2i·28-s + 6·29-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s − 0.755i·7-s − 0.353i·8-s + 0.904·11-s + 0.554i·13-s + 0.534·14-s + 0.250·16-s − 0.727i·17-s + 0.229·19-s + 0.639i·22-s + 1.25i·23-s − 0.392·26-s + 0.377i·28-s + 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4050\)    =    \(2 \cdot 3^{4} \cdot 5^{2}\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(32.3394\)
Root analytic conductor: \(5.68677\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4050} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4050,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.844695765\)
\(L(\frac12)\) \(\approx\) \(1.844695765\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 - 3T + 11T^{2} \)
13 \( 1 - 2iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 - T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 9T + 41T^{2} \)
43 \( 1 + iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 + 12iT - 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + 5iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 - 11iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 12iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 5iT - 97T^{2} \)
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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

−8.443682876602211785625774750278, −7.64020727177535560718338850635, −6.82942766353346685523230801964, −6.66580189105063232319291758141, −5.47335187325497441637113811051, −4.86230117149344949777863060800, −3.92493503189121857420431104568, −3.35991565849462132810090556189, −1.86068554715974495810167109786, −0.74374774707116788588238715102, 0.857266828283145385884564880879, 1.96765964058205846150258867384, 2.81319050715430312872585402958, 3.68700068019899057794926897598, 4.48241112114823945773560563889, 5.35016854750533457278605774730, 6.11115844245026357242835008025, 6.81243767401715786193774743583, 7.88437008125102159658082766725, 8.598757719684644525004190562698

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