Properties

Label 2-4140-5.4-c1-0-19
Degree $2$
Conductor $4140$
Sign $-0.234 - 0.972i$
Analytic cond. $33.0580$
Root an. cond. $5.74961$
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  + (2.17 − 0.523i)5-s + 4.50i·7-s + 4.10·11-s + 4.10i·13-s + 2.26i·17-s − 6.77·19-s + i·23-s + (4.45 − 2.27i)25-s + 4.13·29-s + 1.84·31-s + (2.36 + 9.80i)35-s + 11.1i·37-s − 8.36·41-s − 5.43i·43-s + 0.593i·47-s + ⋯
L(s)  = 1  + (0.972 − 0.234i)5-s + 1.70i·7-s + 1.23·11-s + 1.13i·13-s + 0.549i·17-s − 1.55·19-s + 0.208i·23-s + (0.890 − 0.455i)25-s + 0.768·29-s + 0.330·31-s + (0.398 + 1.65i)35-s + 1.82i·37-s − 1.30·41-s − 0.828i·43-s + 0.0865i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.234 - 0.972i$
Analytic conductor: \(33.0580\)
Root analytic conductor: \(5.74961\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4140} (829, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4140,\ (\ :1/2),\ -0.234 - 0.972i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.179438662\)
\(L(\frac12)\) \(\approx\) \(2.179438662\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-2.17 + 0.523i)T \)
23 \( 1 - iT \)
good7 \( 1 - 4.50iT - 7T^{2} \)
11 \( 1 - 4.10T + 11T^{2} \)
13 \( 1 - 4.10iT - 13T^{2} \)
17 \( 1 - 2.26iT - 17T^{2} \)
19 \( 1 + 6.77T + 19T^{2} \)
29 \( 1 - 4.13T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 - 11.1iT - 37T^{2} \)
41 \( 1 + 8.36T + 41T^{2} \)
43 \( 1 + 5.43iT - 43T^{2} \)
47 \( 1 - 0.593iT - 47T^{2} \)
53 \( 1 - 1.70iT - 53T^{2} \)
59 \( 1 - 6.19T + 59T^{2} \)
61 \( 1 + 11.3T + 61T^{2} \)
67 \( 1 + 5.78iT - 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 - 0.363iT - 73T^{2} \)
79 \( 1 + 1.75T + 79T^{2} \)
83 \( 1 + 9.72iT - 83T^{2} \)
89 \( 1 + 17.2T + 89T^{2} \)
97 \( 1 - 4.38iT - 97T^{2} \)
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

−8.584230725264262250377902459105, −8.398907683361077012022444175663, −6.65107634145747160750100303193, −6.52323653015708853048293426234, −5.81195636712125609113208365294, −4.92815686196758609087193877345, −4.22497717799505504769229366749, −3.02349260295233021903107286889, −2.04507785988943521520266937616, −1.57064466168058140356831009471, 0.60066530396457327524693845423, 1.52938309788626654142120747812, 2.66952635290840115429642714801, 3.67671678333987944508475167212, 4.33222734227682177350682848716, 5.20634842908520667994851773231, 6.22815950395190750672705019168, 6.69260709873358566540238328339, 7.31600539424666295431267189991, 8.217537146787565725727420762699

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