↑

Properties

Label 2-102-51.47-c2-0-8
Degree $2$
Conductor $102$
Sign $0.405 + 0.913i$
Analytic cond. $2.77929$
Root an. cond. $1.66712$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.41·2-s + (2.12 − 2.12i)3-s + 2.00·4-s + (4.94 − 4.94i)5-s + (−3 + 3i)6-s + (−5 + 5i)7-s − 2.82·8-s − 8.99i·9-s + (−7.00 + 7.00i)10-s + (−3.53 − 3.53i)11-s + (4.24 − 4.24i)12-s + 15·13-s + (7.07 − 7.07i)14-s − 20.9i·15-s + 4.00·16-s + (−4.94 − 16.2i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.707 − 0.707i)3-s + 0.500·4-s + (0.989 − 0.989i)5-s + (−0.5 + 0.5i)6-s + (−0.714 + 0.714i)7-s − 0.353·8-s − 0.999i·9-s + (−0.700 + 0.700i)10-s + (−0.321 − 0.321i)11-s + (0.353 − 0.353i)12-s + 1.15·13-s + (0.505 − 0.505i)14-s − 1.39i·15-s + 0.250·16-s + (−0.291 − 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.405 + 0.913i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 102 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.405 + 0.913i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(102\)    =    \(2 \cdot 3 \cdot 17\)
Sign: $0.405 + 0.913i$
Analytic conductor: \(2.77929\)
Root analytic conductor: \(1.66712\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{102} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 102,\ (\ :1),\ 0.405 + 0.913i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.09886 - 0.714353i\)
\(L(\frac12)\) \(\approx\) \(1.09886 - 0.714353i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (-2.12 + 2.12i)T \)
17 \( 1 + (4.94 + 16.2i)T \)
good5 \( 1 + (-4.94 + 4.94i)T - 25iT^{2} \)
7 \( 1 + (5 - 5i)T - 49iT^{2} \)
11 \( 1 + (3.53 + 3.53i)T + 121iT^{2} \)
13 \( 1 - 15T + 169T^{2} \)
19 \( 1 - 5iT - 361T^{2} \)
23 \( 1 + (-14.8 - 14.8i)T + 529iT^{2} \)
29 \( 1 + (28.2 - 28.2i)T - 841iT^{2} \)
31 \( 1 + (-24 - 24i)T + 961iT^{2} \)
37 \( 1 + (15 + 15i)T + 1.36e3iT^{2} \)
41 \( 1 + (-38.8 - 38.8i)T + 1.68e3iT^{2} \)
43 \( 1 - 75iT - 1.84e3T^{2} \)
47 \( 1 + 11.3iT - 2.20e3T^{2} \)
53 \( 1 + 67.8T + 2.80e3T^{2} \)
59 \( 1 - 63.6T + 3.48e3T^{2} \)
61 \( 1 + (-71 + 71i)T - 3.72e3iT^{2} \)
67 \( 1 + 80T + 4.48e3T^{2} \)
71 \( 1 + (-28.2 + 28.2i)T - 5.04e3iT^{2} \)
73 \( 1 + (-80 - 80i)T + 5.32e3iT^{2} \)
79 \( 1 + (11 - 11i)T - 6.24e3iT^{2} \)
83 \( 1 + 124.T + 6.88e3T^{2} \)
89 \( 1 + 21.2iT - 7.92e3T^{2} \)
97 \( 1 + (80 + 80i)T + 9.40e3iT^{2} \)
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

−13.16413246051568461228923734152, −12.69387969420590137881465325215, −11.29343333549993383002621080503, −9.571772537409794473480195161627, −9.108931988421568105722706446159, −8.186428782327615660859119976536, −6.64696842506397714282667681152, −5.57805414377180645613688498533, −2.96884175013973575601260425664, −1.35577251202211552047011528125, 2.35953668257257693411141766632, 3.78525908697981197875014116100, 6.00737550732206080263679791704, 7.12777077591844217081655235600, 8.524598434370390393591518002996, 9.655930444269371025816592530332, 10.38916438302884485705698939595, 10.98508970397503544711540111696, 13.12200638184886733399029956800, 13.78821671985063993846839749105

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