↑

Properties

Label 2-4140-5.4-c1-0-47
Degree $2$
Conductor $4140$
Sign $-0.0684 + 0.997i$
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  + (2.23 + 0.152i)5-s − 4.09i·7-s − 0.777·11-s − 1.67i·13-s − 3.55i·17-s + 4.61·19-s + i·23-s + (4.95 + 0.682i)25-s − 8.24·29-s + 6.43·31-s + (0.626 − 9.13i)35-s + 1.05i·37-s + 5.98·41-s + 3.37i·43-s + 1.24i·47-s + ⋯
L(s)  = 1  + (0.997 + 0.0684i)5-s − 1.54i·7-s − 0.234·11-s − 0.464i·13-s − 0.862i·17-s + 1.05·19-s + 0.208i·23-s + (0.990 + 0.136i)25-s − 1.53·29-s + 1.15·31-s + (0.105 − 1.54i)35-s + 0.173i·37-s + 0.935·41-s + 0.514i·43-s + 0.181i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0684 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0684 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4140\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 23\)
Sign: $-0.0684 + 0.997i$
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: \((2,\ 4140,\ (\ :1/2),\ -0.0684 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.163089615\)
\(L(\frac12)\) \(\approx\) \(2.163089615\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 - 0.152i)T \)
23 \( 1 - iT \)
good7 \( 1 + 4.09iT - 7T^{2} \)
11 \( 1 + 0.777T + 11T^{2} \)
13 \( 1 + 1.67iT - 13T^{2} \)
17 \( 1 + 3.55iT - 17T^{2} \)
19 \( 1 - 4.61T + 19T^{2} \)
29 \( 1 + 8.24T + 29T^{2} \)
31 \( 1 - 6.43T + 31T^{2} \)
37 \( 1 - 1.05iT - 37T^{2} \)
41 \( 1 - 5.98T + 41T^{2} \)
43 \( 1 - 3.37iT - 43T^{2} \)
47 \( 1 - 1.24iT - 47T^{2} \)
53 \( 1 + 6.08iT - 53T^{2} \)
59 \( 1 - 2.37T + 59T^{2} \)
61 \( 1 + 2.12T + 61T^{2} \)
67 \( 1 + 14.7iT - 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 + 15.7iT - 73T^{2} \)
79 \( 1 - 1.63T + 79T^{2} \)
83 \( 1 - 17.0iT - 83T^{2} \)
89 \( 1 + 5.52T + 89T^{2} \)
97 \( 1 + 8.78iT - 97T^{2} \)
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

−7.999550547626362099388427147780, −7.48297080847228771352702349042, −6.85063343013314087521158826403, −6.01946543762877969128497221473, −5.23053538043173244005562136687, −4.54791892136718935372094023689, −3.51498194995924814941390503440, −2.76939773434856299919929797451, −1.55203230953233857379798430917, −0.62038886687056769095146955378, 1.37418850765067376382356631678, 2.28063920708390588932357040801, 2.88737944907081056392015629067, 4.09067203243377340317399740317, 5.14702269227359957126295595931, 5.71278135263167979251473576587, 6.14120257224554110662051411843, 7.08645868581606164647623450696, 7.984084646972609313741299602806, 8.869711089441124836045438110890

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