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Properties

Label 2-666-111.98-c1-0-7
Degree $2$
Conductor $666$
Sign $0.997 + 0.0714i$
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 + (3.66 + 0.320i)5-s + (−1.19 + 0.999i)7-s + (0.965 − 0.258i)8-s + (−1.83 − 3.18i)10-s + (−1.53 + 2.65i)11-s + (4.68 − 2.18i)13-s + (1.50 + 0.402i)14-s + (−0.766 − 0.642i)16-s + (3.89 + 1.81i)17-s + (−3.65 − 2.55i)19-s + (−1.55 + 3.33i)20-s + (3.05 − 0.267i)22-s + (−1.44 + 5.38i)23-s + ⋯
L(s)  = 1  + (−0.405 − 0.579i)2-s + (−0.171 + 0.469i)4-s + (1.63 + 0.143i)5-s + (−0.450 + 0.377i)7-s + (0.341 − 0.0915i)8-s + (−0.581 − 1.00i)10-s + (−0.462 + 0.801i)11-s + (1.29 − 0.605i)13-s + (0.401 + 0.107i)14-s + (−0.191 − 0.160i)16-s + (0.945 + 0.440i)17-s + (−0.837 − 0.586i)19-s + (−0.347 + 0.744i)20-s + (0.651 − 0.0570i)22-s + (−0.300 + 1.12i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0714i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 + 0.0714i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.997 + 0.0714i$
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: \((2,\ 666,\ (\ :1/2),\ 0.997 + 0.0714i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.55292 - 0.0555605i\)
\(L(\frac12)\) \(\approx\) \(1.55292 - 0.0555605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.10 + 3.31i)T \)
good5 \( 1 + (-3.66 - 0.320i)T + (4.92 + 0.868i)T^{2} \)
7 \( 1 + (1.19 - 0.999i)T + (1.21 - 6.89i)T^{2} \)
11 \( 1 + (1.53 - 2.65i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-4.68 + 2.18i)T + (8.35 - 9.95i)T^{2} \)
17 \( 1 + (-3.89 - 1.81i)T + (10.9 + 13.0i)T^{2} \)
19 \( 1 + (3.65 + 2.55i)T + (6.49 + 17.8i)T^{2} \)
23 \( 1 + (1.44 - 5.38i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (-1.43 - 5.35i)T + (-25.1 + 14.5i)T^{2} \)
31 \( 1 + (-0.808 - 0.808i)T + 31iT^{2} \)
41 \( 1 + (-4.40 - 1.60i)T + (31.4 + 26.3i)T^{2} \)
43 \( 1 + (-2.81 + 2.81i)T - 43iT^{2} \)
47 \( 1 + (-6.67 + 3.85i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.78 + 2.12i)T + (-9.20 - 52.1i)T^{2} \)
59 \( 1 + (0.929 + 10.6i)T + (-58.1 + 10.2i)T^{2} \)
61 \( 1 + (2.73 + 5.86i)T + (-39.2 + 46.7i)T^{2} \)
67 \( 1 + (-6.26 - 7.46i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-11.1 + 1.96i)T + (66.7 - 24.2i)T^{2} \)
73 \( 1 + 2.09iT - 73T^{2} \)
79 \( 1 + (-1.21 + 13.8i)T + (-77.7 - 13.7i)T^{2} \)
83 \( 1 + (-2.26 - 6.23i)T + (-63.5 + 53.3i)T^{2} \)
89 \( 1 + (14.6 - 1.28i)T + (87.6 - 15.4i)T^{2} \)
97 \( 1 + (14.4 + 3.88i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.46141512037160129374146564937, −9.695107791312777313322390472617, −9.079570306056151527952820417984, −8.141043167702910780469743263708, −6.90236187474009634465107637447, −5.93729975741712590088684410789, −5.23637195017736311208778862978, −3.60307484550655430361175741680, −2.49474574925409342399167363478, −1.47911902498247134106064971528, 1.11621145299750387991400729783, 2.53803843537838431534469920188, 4.09045980449910228017278275112, 5.52101604817882667001764387166, 6.08788313286003947902415905480, 6.71786846975459772153326352265, 8.097188838757360590588017625815, 8.800876161710643613887500664219, 9.656044536472782210415892855262, 10.32745307667119458471774512897

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