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Properties

Label 2-666-111.17-c1-0-4
Degree $2$
Conductor $666$
Sign $0.903 + 0.429i$
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  + (−0.573 + 0.819i)2-s + (−0.342 − 0.939i)4-s + (−3.25 + 0.284i)5-s + (−0.761 − 0.638i)7-s + (0.965 + 0.258i)8-s + (1.63 − 2.83i)10-s + (2.46 + 4.26i)11-s + (−3.00 − 1.40i)13-s + (0.960 − 0.257i)14-s + (−0.766 + 0.642i)16-s + (−0.884 + 0.412i)17-s + (6.51 − 4.56i)19-s + (1.38 + 2.96i)20-s + (−4.91 − 0.429i)22-s + (−1.73 − 6.46i)23-s + ⋯
L(s)  = 1  + (−0.405 + 0.579i)2-s + (−0.171 − 0.469i)4-s + (−1.45 + 0.127i)5-s + (−0.287 − 0.241i)7-s + (0.341 + 0.0915i)8-s + (0.516 − 0.895i)10-s + (0.743 + 1.28i)11-s + (−0.833 − 0.388i)13-s + (0.256 − 0.0687i)14-s + (−0.191 + 0.160i)16-s + (−0.214 + 0.100i)17-s + (1.49 − 1.04i)19-s + (0.308 + 0.662i)20-s + (−1.04 − 0.0916i)22-s + (−0.361 − 1.34i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.903 + 0.429i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.903 + 0.429i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.903 + 0.429i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.698686 - 0.157674i\)
\(L(\frac12)\) \(\approx\) \(0.698686 - 0.157674i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.573 - 0.819i)T \)
3 \( 1 \)
37 \( 1 + (-2.37 - 5.60i)T \)
good5 \( 1 + (3.25 - 0.284i)T + (4.92 - 0.868i)T^{2} \)
7 \( 1 + (0.761 + 0.638i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-2.46 - 4.26i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.00 + 1.40i)T + (8.35 + 9.95i)T^{2} \)
17 \( 1 + (0.884 - 0.412i)T + (10.9 - 13.0i)T^{2} \)
19 \( 1 + (-6.51 + 4.56i)T + (6.49 - 17.8i)T^{2} \)
23 \( 1 + (1.73 + 6.46i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-1.24 + 4.64i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-7.57 + 7.57i)T - 31iT^{2} \)
41 \( 1 + (-7.43 + 2.70i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (1.60 + 1.60i)T + 43iT^{2} \)
47 \( 1 + (-3.88 - 2.24i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.06 + 8.42i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (0.720 - 8.23i)T + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (3.71 - 7.97i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (1.59 - 1.89i)T + (-11.6 - 65.9i)T^{2} \)
71 \( 1 + (5.07 + 0.894i)T + (66.7 + 24.2i)T^{2} \)
73 \( 1 - 5.03iT - 73T^{2} \)
79 \( 1 + (-0.278 - 3.18i)T + (-77.7 + 13.7i)T^{2} \)
83 \( 1 + (-3.09 + 8.50i)T + (-63.5 - 53.3i)T^{2} \)
89 \( 1 + (5.91 + 0.517i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (-11.4 + 3.07i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.16955595983152807213696210558, −9.650020448223171381491981301246, −8.567135235959343606996136074984, −7.59490730177684338060315931985, −7.23714947677144242372894337093, −6.28117842072093014980664319632, −4.71611260065913843435355536511, −4.20842937148244174874986818922, −2.69629893547749514731847645285, −0.54712928448343228198583327443, 1.12237424468986961176868979566, 3.10136977690793833510052680107, 3.68866242849198404595988976045, 4.85429871258052910201591878328, 6.17801494542483143369937597266, 7.44375960635166222944732063019, 7.951912875261990053582597622968, 8.971504096264490286735924156494, 9.577155114273327096188387629573, 10.75488841049895515156986826504

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