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Properties

Label 2-666-111.98-c1-0-13
Degree $2$
Conductor $666$
Sign $0.917 - 0.397i$
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 + (1.60 + 0.140i)5-s + (1.53 − 1.29i)7-s + (−0.965 + 0.258i)8-s + (0.807 + 1.39i)10-s + (2.64 − 4.57i)11-s + (4.54 − 2.11i)13-s + (1.94 + 0.520i)14-s + (−0.766 − 0.642i)16-s + (−5.07 − 2.36i)17-s + (−0.363 − 0.254i)19-s + (−0.682 + 1.46i)20-s + (5.26 − 0.460i)22-s + (−1.51 + 5.64i)23-s + ⋯
L(s)  = 1  + (0.405 + 0.579i)2-s + (−0.171 + 0.469i)4-s + (0.719 + 0.0629i)5-s + (0.582 − 0.488i)7-s + (−0.341 + 0.0915i)8-s + (0.255 + 0.442i)10-s + (0.797 − 1.38i)11-s + (1.25 − 0.587i)13-s + (0.518 + 0.139i)14-s + (−0.191 − 0.160i)16-s + (−1.23 − 0.574i)17-s + (−0.0833 − 0.0583i)19-s + (−0.152 + 0.327i)20-s + (1.12 − 0.0982i)22-s + (−0.315 + 1.17i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.397i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.917 - 0.397i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.917 - 0.397i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.19610 + 0.455345i\)
\(L(\frac12)\) \(\approx\) \(2.19610 + 0.455345i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.05 - 0.580i)T \)
good5 \( 1 + (-1.60 - 0.140i)T + (4.92 + 0.868i)T^{2} \)
7 \( 1 + (-1.53 + 1.29i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (-2.64 + 4.57i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.54 + 2.11i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (5.07 + 2.36i)T + (10.9 + 13.0i)T^{2} \)
19 \( 1 + (0.363 + 0.254i)T + (6.49 + 17.8i)T^{2} \)
23 \( 1 + (1.51 - 5.64i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.91 - 7.15i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-5.12 - 5.12i)T + 31iT^{2} \)
41 \( 1 + (6.41 + 2.33i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-7.63 + 7.63i)T - 43iT^{2} \)
47 \( 1 + (4.34 - 2.50i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (7.64 - 9.10i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.615 + 7.03i)T + (-58.1 + 10.2i)T^{2} \)
61 \( 1 + (2.86 + 6.14i)T + (-39.2 + 46.7i)T^{2} \)
67 \( 1 + (3.96 + 4.72i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.49 + 0.263i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 - 8.32iT - 73T^{2} \)
79 \( 1 + (1.28 - 14.6i)T + (-77.7 - 13.7i)T^{2} \)
83 \( 1 + (-4.01 - 11.0i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (-0.608 + 0.0532i)T + (87.6 - 15.4i)T^{2} \)
97 \( 1 + (10.0 + 2.68i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.91452220725059644218594116290, −9.512575989818398696700072185753, −8.702094989049313242475059069220, −8.020237004679782455679235952686, −6.78509281540242035832510768952, −6.11997862195577031412884525595, −5.28221747098021808848138468568, −4.09470659987592817940757681654, −3.09994781852700510341549573371, −1.31120314410692083616165421117, 1.64432667647500478986682845600, 2.34665891852775666872476327467, 4.16077695441224458188823795813, 4.60746494129152478702314836668, 6.15252398632920891017384069389, 6.42074675187870982951887562369, 8.037633087817109939441594986333, 8.949073978952302716409244764907, 9.666591199277376639515890363414, 10.47706073524551139610497026620

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