Properties

Label 2-675-75.56-c0-0-1
Degree $2$
Conductor $675$
Sign $0.929 - 0.368i$
Analytic cond. $0.336868$
Root an. cond. $0.580404$
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  + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s + (0.951 + 0.309i)5-s + 0.618·7-s + (−0.587 + 0.809i)8-s + 0.618·10-s + (−0.951 + 0.309i)11-s + (0.363 − 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 + 0.190i)20-s + (−0.5 + 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.224i)28-s + (−0.951 − 1.30i)29-s + ⋯
L(s)  = 1  + (0.587 − 0.190i)2-s + (−0.5 + 0.363i)4-s + (0.951 + 0.309i)5-s + 0.618·7-s + (−0.587 + 0.809i)8-s + 0.618·10-s + (−0.951 + 0.309i)11-s + (0.363 − 0.118i)14-s + (0.587 − 0.809i)17-s + (0.809 + 0.587i)19-s + (−0.587 + 0.190i)20-s + (−0.5 + 0.363i)22-s + (−0.951 + 0.309i)23-s + (0.809 + 0.587i)25-s + (−0.309 + 0.224i)28-s + (−0.951 − 1.30i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(675\)    =    \(3^{3} \cdot 5^{2}\)
Sign: $0.929 - 0.368i$
Analytic conductor: \(0.336868\)
Root analytic conductor: \(0.580404\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{675} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 675,\ (\ :0),\ 0.929 - 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.259963405\)
\(L(\frac12)\) \(\approx\) \(1.259963405\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (-0.951 - 0.309i)T \)
good2 \( 1 + (-0.587 + 0.190i)T + (0.809 - 0.587i)T^{2} \)
7 \( 1 - 0.618T + T^{2} \)
11 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.587 + 0.809i)T + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.951 + 0.309i)T + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.363 - 0.5i)T + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
71 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.951 - 0.309i)T + (0.809 - 0.587i)T^{2} \)
97 \( 1 + (0.809 - 0.587i)T + (0.309 - 0.951i)T^{2} \)
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

−10.81497037841400251153920725475, −9.811462822591195747989444870640, −9.261144389954184560313227687647, −7.975106745476583637766005351509, −7.47558461381834210035221137176, −5.82900670623603928278620804420, −5.39483418429512394676594629622, −4.34399642510628811767073224228, −3.10883424376430230109029503288, −2.06740151735685846216634824003, 1.50814893227143101926670875839, 3.08998362421842038756747751363, 4.44850067275298932576405426655, 5.36651034757042425334124080628, 5.75429391676744977564228600231, 6.93998659307715974545814567733, 8.210246893119073269046367947751, 8.894533032095395275817260810357, 9.993145359804615478825722990817, 10.35635573482981870447576145611

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