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Properties

Label 2-666-111.14-c1-0-6
Degree $2$
Conductor $666$
Sign $-0.200 + 0.979i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.200 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.200 + 0.979i$
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: \((2,\ 666,\ (\ :1/2),\ -0.200 + 0.979i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.975006 - 1.19496i\)
\(L(\frac12)\) \(\approx\) \(0.975006 - 1.19496i\)
\(L(\frac{3}{2})\) 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

−10.16640178908842972476069818883, −9.267499692998938293984522458715, −8.720843840711833018588999467457, −8.191156442233113839044465788357, −6.52897432267769697211291300420, −5.66044447749669263183459575233, −4.50867400796500222907956540066, −3.79500040730989043570827445766, −1.99495815044823884045262345047, −1.03412299896558318611596580019, 1.59551931832647791367798104305, 3.32007312215929481981711908625, 4.13305128061001138629158340862, 5.85430337269808468075217034119, 6.30261929821711036783742254049, 7.09242621592012087340183070463, 8.062327536319307150692906964659, 8.938400613705980116293424199673, 9.971861232235404984582057468382, 10.70811600348675740690534102785

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