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Properties

Label 2-666-111.8-c1-0-2
Degree $2$
Conductor $666$
Sign $-0.999 + 0.0263i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.653 + 2.43i)5-s + (−1.76 + 3.05i)7-s + (0.707 − 0.707i)8-s − 2.52·10-s + 2.53·11-s + (−2.84 + 0.762i)13-s + (−2.49 − 2.49i)14-s + (0.500 + 0.866i)16-s + (−1.73 − 0.463i)17-s + (−3.53 + 0.946i)19-s + (0.653 − 2.43i)20-s + (−0.656 + 2.44i)22-s + (3.15 − 3.15i)23-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.433 − 0.249i)4-s + (0.292 + 1.09i)5-s + (−0.666 + 1.15i)7-s + (0.249 − 0.249i)8-s − 0.798·10-s + 0.764·11-s + (−0.788 + 0.211i)13-s + (−0.666 − 0.666i)14-s + (0.125 + 0.216i)16-s + (−0.419 − 0.112i)17-s + (−0.809 + 0.217i)19-s + (0.146 − 0.545i)20-s + (−0.139 + 0.521i)22-s + (0.658 − 0.658i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0263i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.999 + 0.0263i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.999 + 0.0263i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0118617 - 0.898466i\)
\(L(\frac12)\) \(\approx\) \(0.0118617 - 0.898466i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 \)
37 \( 1 + (6.08 + 0.00578i)T \)
good5 \( 1 + (-0.653 - 2.43i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.76 - 3.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 - 2.53T + 11T^{2} \)
13 \( 1 + (2.84 - 0.762i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (1.73 + 0.463i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (3.53 - 0.946i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.15 + 3.15i)T - 23iT^{2} \)
29 \( 1 + (-0.0393 - 0.0393i)T + 29iT^{2} \)
31 \( 1 + (3.02 - 3.02i)T - 31iT^{2} \)
41 \( 1 + (4.63 - 8.02i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.234 - 0.234i)T + 43iT^{2} \)
47 \( 1 + 4.39iT - 47T^{2} \)
53 \( 1 + (3.09 - 1.78i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-9.18 - 2.46i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.55 - 9.52i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (13.5 + 7.80i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.56 - 2.05i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.09iT - 73T^{2} \)
79 \( 1 + (-2.84 + 0.762i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-7.86 + 4.54i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (3.78 - 14.1i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.841 + 0.841i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.74328864269348509610880684778, −9.968233911068869728109010682362, −9.119545359275581369403696536020, −8.510016468213430438989402274229, −7.08680468507811356278857495797, −6.62646421236296831140167210255, −5.86807253297327691394663495384, −4.71328638711626772012022145895, −3.26978466504667234405815506462, −2.20239562393055782254661434296, 0.49972148034612359512639906901, 1.83285663961598552463903659445, 3.45215463009681861433789230981, 4.34784544804669600412535286205, 5.26547100195245825213034725760, 6.63269226573607291272462236901, 7.48237204921658893115539530685, 8.702577164874303712559065206049, 9.276420428212679866678702866928, 10.06241878530078855034190121789

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