↑

Properties

Label 2-828-92.91-c1-0-51
Degree $2$
Conductor $828$
Sign $-0.861 - 0.507i$
Analytic cond. $6.61161$
Root an. cond. $2.57130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.279 − 1.38i)2-s + (−1.84 − 0.775i)4-s − 1.27i·5-s + 2.49·7-s + (−1.58 + 2.33i)8-s + (−1.76 − 0.356i)10-s − 5.92·11-s − 2.46·13-s + (0.696 − 3.45i)14-s + (2.79 + 2.85i)16-s − 6.74i·17-s − 2.53·19-s + (−0.988 + 2.35i)20-s + (−1.65 + 8.20i)22-s + (−4.75 − 0.642i)23-s + ⋯
L(s)  = 1  + (0.197 − 0.980i)2-s + (−0.921 − 0.387i)4-s − 0.570i·5-s + 0.941·7-s + (−0.562 + 0.827i)8-s + (−0.559 − 0.112i)10-s − 1.78·11-s − 0.684·13-s + (0.186 − 0.923i)14-s + (0.699 + 0.714i)16-s − 1.63i·17-s − 0.582·19-s + (−0.221 + 0.525i)20-s + (−0.352 + 1.75i)22-s + (−0.990 − 0.133i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 828 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(828\)    =    \(2^{2} \cdot 3^{2} \cdot 23\)
Sign: $-0.861 - 0.507i$
Analytic conductor: \(6.61161\)
Root analytic conductor: \(2.57130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{828} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 828,\ (\ :1/2),\ -0.861 - 0.507i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.201509 + 0.739080i\)
\(L(\frac12)\) \(\approx\) \(0.201509 + 0.739080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.279 + 1.38i)T \)
3 \( 1 \)
23 \( 1 + (4.75 + 0.642i)T \)
good5 \( 1 + 1.27iT - 5T^{2} \)
7 \( 1 - 2.49T + 7T^{2} \)
11 \( 1 + 5.92T + 11T^{2} \)
13 \( 1 + 2.46T + 13T^{2} \)
17 \( 1 + 6.74iT - 17T^{2} \)
19 \( 1 + 2.53T + 19T^{2} \)
29 \( 1 + 5.45T + 29T^{2} \)
31 \( 1 + 3.02iT - 31T^{2} \)
37 \( 1 + 4.92iT - 37T^{2} \)
41 \( 1 - 9.26T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 - 9.93iT - 47T^{2} \)
53 \( 1 - 2.08iT - 53T^{2} \)
59 \( 1 - 7.43iT - 59T^{2} \)
61 \( 1 + 2.14iT - 61T^{2} \)
67 \( 1 - 3.96T + 67T^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 4.45T + 73T^{2} \)
79 \( 1 - 3.65T + 79T^{2} \)
83 \( 1 - 3.03T + 83T^{2} \)
89 \( 1 + 5.81iT - 89T^{2} \)
97 \( 1 - 4.52iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.822670280924293670699637703330, −9.079307580583448213504476011726, −8.075973108318125256597194318095, −7.51762483077752435818228450138, −5.73439834625855353656064285096, −4.96953149347678134240570115446, −4.46057343914974544245163300654, −2.87074365479628340770688911932, −2.01990364053980621221588976065, −0.33303893203844001979588918530, 2.18239854242969755924763721580, 3.56513175405765792765661738575, 4.73516369509017725668850031399, 5.42881746039036009110300427316, 6.38399243188951561242790855077, 7.39777676915482987333937241012, 8.071948028624837227321197326423, 8.568130313652967382487418692217, 10.01328982622417149495462779462, 10.48349008159350464741950429602

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