↑

Properties

Label 2-55-5.4-c1-0-1
Degree $2$
Conductor $55$
Sign $0.977 - 0.210i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
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  − 0.792i·2-s + 2.52i·3-s + 1.37·4-s + (−2.18 + 0.469i)5-s + 2·6-s − 3.46i·7-s − 2.67i·8-s − 3.37·9-s + (0.372 + 1.73i)10-s − 11-s + 3.46i·12-s − 2.74·14-s + (−1.18 − 5.51i)15-s + 0.627·16-s + 5.04i·17-s + 2.67i·18-s + ⋯
L(s)  = 1  − 0.560i·2-s + 1.45i·3-s + 0.686·4-s + (−0.977 + 0.210i)5-s + 0.816·6-s − 1.30i·7-s − 0.944i·8-s − 1.12·9-s + (0.117 + 0.547i)10-s − 0.301·11-s + 0.999i·12-s − 0.733·14-s + (−0.306 − 1.42i)15-s + 0.156·16-s + 1.22i·17-s + 0.629i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.977 - 0.210i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.977 - 0.210i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (34, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.977 - 0.210i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.860315 + 0.0914128i\)
\(L(\frac12)\) \(\approx\) \(0.860315 + 0.0914128i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.18 - 0.469i)T \)
11 \( 1 + T \)
good2 \( 1 + 0.792iT - 2T^{2} \)
3 \( 1 - 2.52iT - 3T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 5.04iT - 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + 2.52iT - 23T^{2} \)
29 \( 1 - 2.74T + 29T^{2} \)
31 \( 1 + 2.37T + 31T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + 2.74T + 41T^{2} \)
43 \( 1 - 3.46iT - 43T^{2} \)
47 \( 1 + 6.63iT - 47T^{2} \)
53 \( 1 + 3.16iT - 53T^{2} \)
59 \( 1 - 1.62T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 - 0.644iT - 67T^{2} \)
71 \( 1 - 7.11T + 71T^{2} \)
73 \( 1 + 6.92iT - 73T^{2} \)
79 \( 1 + 12.7T + 79T^{2} \)
83 \( 1 + 6.63iT - 83T^{2} \)
89 \( 1 - 4.37T + 89T^{2} \)
97 \( 1 - 4.10iT - 97T^{2} \)
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

−15.39752858295455827622015830390, −14.69536127088178497001264221151, −12.92369122918368406875878966078, −11.51323317728160153252196799477, −10.55003680888481659010168730285, −10.21784533693697580003851458916, −8.261424559393711865296506807208, −6.76134604263451556055217846691, −4.38975127802683186965272439597, −3.47984059806640897081261858555, 2.45217711766523776713592093680, 5.52923801730463138407263999870, 6.85437210822826851515912330194, 7.75464132179680785468636652988, 8.752358483478454298537765325143, 11.23905208913192929239537573227, 12.04501127844926345527565061308, 12.77160542675294909516894765829, 14.32625908956387078472543558768, 15.45163900919693144083861587179

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