↑

Properties

Label 2-3871-553.473-c0-0-18
Degree $2$
Conductor $3871$
Sign $-0.266 - 0.963i$
Analytic cond. $1.93188$
Root an. cond. $1.38992$
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  + (−0.309 − 0.535i)2-s + (0.309 − 0.535i)4-s + (−0.809 − 1.40i)5-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (−0.309 + 0.535i)11-s − 0.618·13-s + (−0.309 + 0.535i)18-s + (−0.809 − 1.40i)19-s − 20-s + 0.381·22-s + (0.809 + 1.40i)23-s + (−0.809 + 1.40i)25-s + (0.190 + 0.330i)26-s + ⋯
L(s)  = 1  + (−0.309 − 0.535i)2-s + (0.309 − 0.535i)4-s + (−0.809 − 1.40i)5-s − 0.999·8-s + (−0.5 − 0.866i)9-s + (−0.499 + 0.866i)10-s + (−0.309 + 0.535i)11-s − 0.618·13-s + (−0.309 + 0.535i)18-s + (−0.809 − 1.40i)19-s − 20-s + 0.381·22-s + (0.809 + 1.40i)23-s + (−0.809 + 1.40i)25-s + (0.190 + 0.330i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $-0.266 - 0.963i$
Analytic conductor: \(1.93188\)
Root analytic conductor: \(1.38992\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (3791, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3871,\ (\ :0),\ -0.266 - 0.963i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3786963973\)
\(L(\frac12)\) \(\approx\) \(0.3786963973\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
79 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (0.309 + 0.535i)T + (-0.5 + 0.866i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.309 - 0.535i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 0.618T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.809 - 1.40i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.309 + 0.535i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.809 + 1.40i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.809 - 1.40i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + 2T + T^{2} \)
89 \( 1 + (-0.309 - 0.535i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - 1.61T + T^{2} \)
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

−8.389838137886710598404954323366, −7.44708210242857686650001052624, −6.75994499959682513223363614008, −5.76118951508807168795356226381, −5.05529303055441624836779201933, −4.40487684556223519659638129469, −3.34385537117018699876328011287, −2.39988388493618516617804625176, −1.24002457103661142995614642660, −0.24026788780695768593603024564, 2.31881791019540299780859328311, 2.88163625277277928170101143733, 3.61602656222504445168631688278, 4.60639946265447629051009736881, 5.77354241337191112298032906464, 6.42319534473018692366179101018, 7.13676729455009193951637485220, 7.63843852899754848100177524609, 8.355850205794731513201754100573, 8.666168806200892129870320672184

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