↑

Properties

Label 2-1200-48.29-c0-0-1
Degree $2$
Conductor $1200$
Sign $-0.382 + 0.923i$
Analytic cond. $0.598878$
Root an. cond. $0.773872$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−0.707 − 0.707i)12-s − 1.00·16-s + 1.41i·17-s + (−0.707 − 0.707i)18-s + (1 + i)19-s − 1.41·23-s − 1.00·24-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (0.707 − 0.707i)3-s − 1.00i·4-s − 1.00i·6-s + (−0.707 − 0.707i)8-s − 1.00i·9-s + (−0.707 − 0.707i)12-s − 1.00·16-s + 1.41i·17-s + (−0.707 − 0.707i)18-s + (1 + i)19-s − 1.41·23-s − 1.00·24-s + (−0.707 − 0.707i)27-s + (−0.707 + 0.707i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1200\)    =    \(2^{4} \cdot 3 \cdot 5^{2}\)
Sign: $-0.382 + 0.923i$
Analytic conductor: \(0.598878\)
Root analytic conductor: \(0.773872\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1200} (701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1200,\ (\ :0),\ -0.382 + 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.817400993\)
\(L(\frac12)\) \(\approx\) \(1.817400993\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 - T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - 1.41iT - T^{2} \)
19 \( 1 + (-1 - i)T + iT^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 - iT^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + 1.41iT - T^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (-1 - i)T + iT^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + T^{2} \)
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

−9.941308522609251798665432272118, −8.873904816791285466931594360543, −8.123330196420296618394460095524, −7.17941249348476041711943969770, −6.16401463795883761344139401552, −5.57226206177328113976871753949, −4.08158034082665512924573218032, −3.52839064144937048498977292805, −2.32311407062823274440728446077, −1.41653257128971909545445554546, 2.44930427984677945439635248465, 3.24688955565716854955685822371, 4.27611281541011731829627389402, 4.99501247829726113877168130486, 5.82370807001801801500211143230, 7.05560212437867572219302770829, 7.63820609103880437956876881908, 8.503495125939194254862027939810, 9.301993950216402422605559432622, 9.911921954395199987151131381259

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