↑

Properties

Label 2-605-55.32-c1-0-11
Degree $2$
Conductor $605$
Sign $0.553 - 0.832i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + (1.03 − 1.03i)2-s + (0.588 − 0.588i)3-s − 0.149i·4-s + (−2.18 + 0.465i)5-s − 1.22i·6-s + (−2.98 + 2.98i)7-s + (1.91 + 1.91i)8-s + 2.30i·9-s + (−1.78 + 2.74i)10-s + (−0.0877 − 0.0877i)12-s + (0.654 + 0.654i)13-s + 6.17i·14-s + (−1.01 + 1.56i)15-s + 4.27·16-s + (−1.36 + 1.36i)17-s + (2.39 + 2.39i)18-s + ⋯
L(s)  = 1  + (0.732 − 0.732i)2-s + (0.339 − 0.339i)3-s − 0.0745i·4-s + (−0.978 + 0.208i)5-s − 0.498i·6-s + (−1.12 + 1.12i)7-s + (0.678 + 0.678i)8-s + 0.768i·9-s + (−0.564 + 0.869i)10-s + (−0.0253 − 0.0253i)12-s + (0.181 + 0.181i)13-s + 1.65i·14-s + (−0.261 + 0.403i)15-s + 1.06·16-s + (−0.330 + 0.330i)17-s + (0.563 + 0.563i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.553 - 0.832i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33958 + 0.718110i\)
\(L(\frac12)\) \(\approx\) \(1.33958 + 0.718110i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 - 0.465i)T \)
11 \( 1 \)
good2 \( 1 + (-1.03 + 1.03i)T - 2iT^{2} \)
3 \( 1 + (-0.588 + 0.588i)T - 3iT^{2} \)
7 \( 1 + (2.98 - 2.98i)T - 7iT^{2} \)
13 \( 1 + (-0.654 - 0.654i)T + 13iT^{2} \)
17 \( 1 + (1.36 - 1.36i)T - 17iT^{2} \)
19 \( 1 + 5.43T + 19T^{2} \)
23 \( 1 + (-1.95 + 1.95i)T - 23iT^{2} \)
29 \( 1 - 1.00T + 29T^{2} \)
31 \( 1 - 0.423T + 31T^{2} \)
37 \( 1 + (-3.48 - 3.48i)T + 37iT^{2} \)
41 \( 1 - 0.577iT - 41T^{2} \)
43 \( 1 + (5.05 + 5.05i)T + 43iT^{2} \)
47 \( 1 + (-0.841 - 0.841i)T + 47iT^{2} \)
53 \( 1 + (-6.42 + 6.42i)T - 53iT^{2} \)
59 \( 1 - 9.30iT - 59T^{2} \)
61 \( 1 - 7.78iT - 61T^{2} \)
67 \( 1 + (3.05 + 3.05i)T + 67iT^{2} \)
71 \( 1 - 8.59T + 71T^{2} \)
73 \( 1 + (3.86 + 3.86i)T + 73iT^{2} \)
79 \( 1 - 2.28T + 79T^{2} \)
83 \( 1 + (-1.40 - 1.40i)T + 83iT^{2} \)
89 \( 1 - 13.9iT - 89T^{2} \)
97 \( 1 + (2.35 + 2.35i)T + 97iT^{2} \)
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

−11.00178163400863405500963987322, −10.24184043325071469639312172902, −8.745317440588938910815978677402, −8.359645793575491602975079013224, −7.23281996131580414045316840821, −6.25810482482644953542888110185, −4.95805566184742461072479641391, −3.95720728718458184233521346978, −2.94701943989845213805512043570, −2.22078381366618819831681475989, 0.63486642036080431356678800275, 3.30511227185466298578637682810, 3.98053928187805318262153067462, 4.69810379334498678626072890785, 6.17043262457753712612708472029, 6.81109029032832067856562226215, 7.56113868159428466353339617278, 8.710425656014966399372944270464, 9.677622316904507413129189547010, 10.42509757233389549556794280445

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