↑

Properties

Label 2-666-111.56-c1-0-1
Degree $2$
Conductor $666$
Sign $-0.740 + 0.672i$
Analytic cond. $5.31803$
Root an. cond. $2.30608$
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  + (0.0871 + 0.996i)2-s + (−0.984 + 0.173i)4-s + (−2.82 + 1.31i)5-s + (1.97 + 0.718i)7-s + (−0.258 − 0.965i)8-s + (−1.55 − 2.70i)10-s + (−1.33 + 2.31i)11-s + (−0.527 + 0.753i)13-s + (−0.543 + 2.02i)14-s + (0.939 − 0.342i)16-s + (−2.87 − 4.11i)17-s + (−2.92 − 0.256i)19-s + (2.55 − 1.78i)20-s + (−2.42 − 1.13i)22-s + (−2.70 − 0.725i)23-s + ⋯
L(s)  = 1  + (0.0616 + 0.704i)2-s + (−0.492 + 0.0868i)4-s + (−1.26 + 0.589i)5-s + (0.745 + 0.271i)7-s + (−0.0915 − 0.341i)8-s + (−0.493 − 0.853i)10-s + (−0.403 + 0.699i)11-s + (−0.146 + 0.208i)13-s + (−0.145 + 0.541i)14-s + (0.234 − 0.0855i)16-s + (−0.698 − 0.997i)17-s + (−0.671 − 0.0587i)19-s + (0.571 − 0.399i)20-s + (−0.517 − 0.241i)22-s + (−0.564 − 0.151i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.740 + 0.672i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (611, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.740 + 0.672i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.117194 - 0.303201i\)
\(L(\frac12)\) \(\approx\) \(0.117194 - 0.303201i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0871 - 0.996i)T \)
3 \( 1 \)
37 \( 1 + (1.40 + 5.91i)T \)
good5 \( 1 + (2.82 - 1.31i)T + (3.21 - 3.83i)T^{2} \)
7 \( 1 + (-1.97 - 0.718i)T + (5.36 + 4.49i)T^{2} \)
11 \( 1 + (1.33 - 2.31i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.527 - 0.753i)T + (-4.44 - 12.2i)T^{2} \)
17 \( 1 + (2.87 + 4.11i)T + (-5.81 + 15.9i)T^{2} \)
19 \( 1 + (2.92 + 0.256i)T + (18.7 + 3.29i)T^{2} \)
23 \( 1 + (2.70 + 0.725i)T + (19.9 + 11.5i)T^{2} \)
29 \( 1 + (4.95 - 1.32i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (1.88 - 1.88i)T - 31iT^{2} \)
41 \( 1 + (0.366 + 2.08i)T + (-38.5 + 14.0i)T^{2} \)
43 \( 1 + (6.00 + 6.00i)T + 43iT^{2} \)
47 \( 1 + (-5.03 + 2.90i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.29 - 6.30i)T + (-40.6 + 34.0i)T^{2} \)
59 \( 1 + (5.91 - 12.6i)T + (-37.9 - 45.1i)T^{2} \)
61 \( 1 + (-5.22 - 3.66i)T + (20.8 + 57.3i)T^{2} \)
67 \( 1 + (2.91 - 8.01i)T + (-51.3 - 43.0i)T^{2} \)
71 \( 1 + (-3.95 - 4.70i)T + (-12.3 + 69.9i)T^{2} \)
73 \( 1 - 9.01iT - 73T^{2} \)
79 \( 1 + (-0.222 - 0.477i)T + (-50.7 + 60.5i)T^{2} \)
83 \( 1 + (3.14 + 0.553i)T + (77.9 + 28.3i)T^{2} \)
89 \( 1 + (-2.30 - 1.07i)T + (57.2 + 68.1i)T^{2} \)
97 \( 1 + (3.13 - 11.7i)T + (-84.0 - 48.5i)T^{2} \)
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

−11.12620735763390057214970699123, −10.25217033407604936111615223508, −9.017495721885232813874317800747, −8.320724998767581701447164495620, −7.24556001745763448160219669676, −7.12846411130170043832982953812, −5.61120163360672872123298717591, −4.62058149961256132880186097259, −3.83421047545505717167619936392, −2.33196521301879739669824707380, 0.16643199921917436885968362401, 1.79398027183848093873137138886, 3.41975604955462289404149030773, 4.25446504996079934754741999424, 5.03320230967826684259183378393, 6.31552993813819425447814294781, 7.916358678979085974837991554957, 8.082194113121738774390983175525, 9.022155441638639541422498336796, 10.21457818122661954884968154619

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