↑

Properties

Label 2-3871-553.394-c0-0-15
Degree $2$
Conductor $3871$
Sign $0.991 + 0.126i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3871 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3871\)    =    \(7^{2} \cdot 79\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(1.93188\)
Root analytic conductor: \(1.38992\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3871} (1500, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3871,\ (\ :0),\ 0.991 + 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.289096914\)
\(L(\frac12)\) \(\approx\) \(1.289096914\)
\(L(1)\) 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

−8.452533799370787078468313463726, −8.252148066896280025931568040250, −7.21665033460470023639852192112, −6.46497320284720858757453451564, −5.56644277039579083052077003886, −5.18541923637781256193198440152, −4.23456501180263449862779518819, −2.94681527158056944418307728554, −2.27543089016608553645157594398, −0.851464515418252297395622892466, 1.35545480193398473659176646527, 2.15581877413023965865285525504, 3.23856142694820928575502665512, 3.46423104129680499569693548471, 5.23377728037433428123523856073, 5.84942517394790515761329877924, 6.39901640121993653424562235441, 7.05601406261659140843951087419, 7.86732822038267579109455548629, 9.060620870398512179547869537435

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