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Properties

Label 2-666-111.14-c1-0-1
Degree $2$
Conductor $666$
Sign $-0.990 + 0.137i$
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 + (−0.896 + 3.34i)5-s + (0.5 + 0.866i)7-s + (−0.707 − 0.707i)8-s − 3.46·10-s − 3.86·11-s + (5.59 + 1.5i)13-s + (−0.707 + 0.707i)14-s + (0.500 − 0.866i)16-s + (−2.63 + 0.707i)17-s + (−6.09 − 1.63i)19-s + (−0.896 − 3.34i)20-s + (−0.999 − 3.73i)22-s + (1.93 + 1.93i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.400 + 1.49i)5-s + (0.188 + 0.327i)7-s + (−0.249 − 0.249i)8-s − 1.09·10-s − 1.16·11-s + (1.55 + 0.416i)13-s + (−0.188 + 0.188i)14-s + (0.125 − 0.216i)16-s + (−0.640 + 0.171i)17-s + (−1.39 − 0.374i)19-s + (−0.200 − 0.748i)20-s + (−0.213 − 0.795i)22-s + (0.402 + 0.402i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.137i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.990 + 0.137i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.990 + 0.137i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0698017 - 1.01151i\)
\(L(\frac12)\) \(\approx\) \(0.0698017 - 1.01151i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6 + i)T \)
good5 \( 1 + (0.896 - 3.34i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 3.86T + 11T^{2} \)
13 \( 1 + (-5.59 - 1.5i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (2.63 - 0.707i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.09 + 1.63i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.93 - 1.93i)T + 23iT^{2} \)
29 \( 1 + (0.138 - 0.138i)T - 29iT^{2} \)
31 \( 1 + (3.63 + 3.63i)T + 31iT^{2} \)
41 \( 1 + (1.03 + 1.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.56 - 7.56i)T - 43iT^{2} \)
47 \( 1 - 8.76iT - 47T^{2} \)
53 \( 1 + (-2.44 - 1.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.57 - 1.22i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.633 - 2.36i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.59 - 1.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.44 - 1.41i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 15.3iT - 73T^{2} \)
79 \( 1 + (0.133 + 0.0358i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-13.9 - 8.05i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (2.17 + 8.10i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-11.0 + 11.0i)T - 97iT^{2} \)
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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

−11.04768442285347594014808311615, −10.28786451057483695620188409378, −8.995718167691601517022194741825, −8.207437472923015613701448743126, −7.38185777362127106213009694553, −6.48513869037990736051018915257, −5.89603449647351372432353324321, −4.51128887522383665968510426003, −3.48883971186609729435696722966, −2.37583210379279927445094346100, 0.50387868570040493431652372658, 1.87291860672408337644139976429, 3.49332820934231172033054489156, 4.47403349640993613708133595439, 5.17507848269194699355866989651, 6.25581743216732412797227538493, 7.79278274165902391209832131071, 8.567132883987713441634548157434, 8.967456157890586603770582551216, 10.39470589260601629041639591734

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