↑

Properties

Label 2-1224-8.5-c1-0-22
Degree $2$
Conductor $1224$
Sign $0.676 + 0.736i$
Analytic cond. $9.77368$
Root an. cond. $3.12629$
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  + (−0.346 − 1.37i)2-s + (−1.75 + 0.951i)4-s − 1.24i·5-s − 3.59·7-s + (1.91 + 2.08i)8-s + (−1.71 + 0.432i)10-s + 1.76i·11-s + 1.45i·13-s + (1.24 + 4.93i)14-s + (2.19 − 3.34i)16-s + 17-s + 3.75i·19-s + (1.18 + 2.19i)20-s + (2.42 − 0.612i)22-s + 4.84·23-s + ⋯
L(s)  = 1  + (−0.245 − 0.969i)2-s + (−0.879 + 0.475i)4-s − 0.558i·5-s − 1.35·7-s + (0.676 + 0.736i)8-s + (−0.541 + 0.136i)10-s + 0.532i·11-s + 0.403i·13-s + (0.333 + 1.31i)14-s + (0.547 − 0.836i)16-s + 0.242·17-s + 0.860i·19-s + (0.265 + 0.490i)20-s + (0.516 − 0.130i)22-s + 1.01·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.676 + 0.736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1224\)    =    \(2^{3} \cdot 3^{2} \cdot 17\)
Sign: $0.676 + 0.736i$
Analytic conductor: \(9.77368\)
Root analytic conductor: \(3.12629\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1224} (613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1224,\ (\ :1/2),\ 0.676 + 0.736i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.024417258\)
\(L(\frac12)\) \(\approx\) \(1.024417258\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.346 + 1.37i)T \)
3 \( 1 \)
17 \( 1 - T \)
good5 \( 1 + 1.24iT - 5T^{2} \)
7 \( 1 + 3.59T + 7T^{2} \)
11 \( 1 - 1.76iT - 11T^{2} \)
13 \( 1 - 1.45iT - 13T^{2} \)
19 \( 1 - 3.75iT - 19T^{2} \)
23 \( 1 - 4.84T + 23T^{2} \)
29 \( 1 + 7.54iT - 29T^{2} \)
31 \( 1 + 1.06T + 31T^{2} \)
37 \( 1 - 4.97iT - 37T^{2} \)
41 \( 1 - 6.51T + 41T^{2} \)
43 \( 1 + 0.946iT - 43T^{2} \)
47 \( 1 - 3.46T + 47T^{2} \)
53 \( 1 + 6.03iT - 53T^{2} \)
59 \( 1 - 6.30iT - 59T^{2} \)
61 \( 1 + 4.77iT - 61T^{2} \)
67 \( 1 - 11.6iT - 67T^{2} \)
71 \( 1 - 8.69T + 71T^{2} \)
73 \( 1 - 14.6T + 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 + 11.5iT - 83T^{2} \)
89 \( 1 + 7.56T + 89T^{2} \)
97 \( 1 - 7.32T + 97T^{2} \)
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

−9.668540619133072335007267951394, −9.105729073626433502560746546686, −8.241528549053139495493264405490, −7.26877890734085025962345087424, −6.26486924029069905424570984084, −5.16229873515260996451149424190, −4.19698434610901721258132592209, −3.33830703556564641968264915314, −2.28848111587801542122097619031, −0.868948022525394123319433110421, 0.70197278102131051777393659550, 2.87002166344696793564199683185, 3.65124242472671573375678382816, 4.98852151870066186116440428126, 5.82903993301798597417925153886, 6.72070933491219345674870079468, 7.08281337070766631774232550687, 8.130520178469692360814958262730, 9.186769942018635118304705061685, 9.429148616796200465252141979655

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