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Properties

Label 2-666-333.175-c1-0-15
Degree $2$
Conductor $666$
Sign $0.496 + 0.867i$
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  − i·2-s + (−1.70 − 0.276i)3-s − 4-s − 0.372i·5-s + (−0.276 + 1.70i)6-s + (0.0210 − 0.0364i)7-s + i·8-s + (2.84 + 0.944i)9-s − 0.372·10-s + (1.29 + 2.23i)11-s + (1.70 + 0.276i)12-s + 2.23i·13-s + (−0.0364 − 0.0210i)14-s + (−0.103 + 0.637i)15-s + 16-s + (−2.25 − 1.29i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.987 − 0.159i)3-s − 0.5·4-s − 0.166i·5-s + (−0.112 + 0.698i)6-s + (0.00795 − 0.0137i)7-s + 0.353i·8-s + (0.949 + 0.314i)9-s − 0.117·10-s + (0.389 + 0.673i)11-s + (0.493 + 0.0797i)12-s + 0.619i·13-s + (−0.00974 − 0.00562i)14-s + (−0.0266 + 0.164i)15-s + 0.250·16-s + (−0.546 − 0.315i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.496 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.496 + 0.867i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (175, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.496 + 0.867i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.908760 - 0.526886i\)
\(L(\frac12)\) \(\approx\) \(0.908760 - 0.526886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (1.70 + 0.276i)T \)
37 \( 1 + (-5.51 + 2.57i)T \)
good5 \( 1 + 0.372iT - 5T^{2} \)
7 \( 1 + (-0.0210 + 0.0364i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.29 - 2.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.23iT - 13T^{2} \)
17 \( 1 + (2.25 + 1.29i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.14 + 2.96i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.66 - 2.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.672 - 0.388i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.09 + 4.67i)T + (15.5 + 26.8i)T^{2} \)
41 \( 1 - 7.23T + 41T^{2} \)
43 \( 1 + (-9.57 + 5.53i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.02 + 8.70i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.61 + 6.25i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (10.8 - 6.26i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-8.09 - 4.67i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 - 2.32T + 67T^{2} \)
71 \( 1 + (0.342 + 0.592i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 13.1T + 73T^{2} \)
79 \( 1 + (3.39 + 1.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.36T + 83T^{2} \)
89 \( 1 + (-5.63 - 3.25i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (5.13 - 2.96i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.65131683254038845189077007602, −9.432364288782974635296613361056, −9.158912775388099835737206330655, −7.49454894410036359157828037320, −6.92334496122409464808224796316, −5.64811434269783597028520175648, −4.81708980309305504111912304032, −3.92589927636192861722376009276, −2.29175619698618561560109241328, −0.930331673297629627009909713512, 0.995877429210210372531013677853, 3.25749072087566390624396794660, 4.44255648683317922357995320915, 5.43854337083546682246872675528, 6.11412110181341376953103998339, 6.99483113512066575289177833046, 7.84628217895808573406542739047, 8.978851014214344369486368959240, 9.738463909026458732054429347600, 10.86243270735391884082823574647

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