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Properties

Label 2-666-111.17-c1-0-5
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: $\chi_{666} (17, \cdot )$
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

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

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