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Properties

Label 2-666-37.10-c1-0-10
Degree $2$
Conductor $666$
Sign $-0.449 + 0.893i$
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  + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−1.18 − 2.05i)5-s + (0.686 + 1.18i)7-s + 0.999·8-s + 2.37·10-s − 2·11-s + (−0.686 − 1.18i)13-s − 1.37·14-s + (−0.5 + 0.866i)16-s + (−0.813 + 1.40i)17-s + (−2.37 − 4.10i)19-s + (−1.18 + 2.05i)20-s + (1 − 1.73i)22-s + (−0.313 + 0.543i)25-s + 1.37·26-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.530 − 0.918i)5-s + (0.259 + 0.449i)7-s + 0.353·8-s + 0.750·10-s − 0.603·11-s + (−0.190 − 0.329i)13-s − 0.366·14-s + (−0.125 + 0.216i)16-s + (−0.197 + 0.341i)17-s + (−0.544 − 0.942i)19-s + (−0.265 + 0.459i)20-s + (0.213 − 0.369i)22-s + (−0.0627 + 0.108i)25-s + 0.269·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.449 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.449 + 0.893i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.449 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.221177 - 0.358810i\)
\(L(\frac12)\) \(\approx\) \(0.221177 - 0.358810i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 \)
37 \( 1 + (2.55 + 5.51i)T \)
good5 \( 1 + (1.18 + 2.05i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.686 - 1.18i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + (0.686 + 1.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.813 - 1.40i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.37 + 4.10i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 4.37T + 29T^{2} \)
31 \( 1 + 9.37T + 31T^{2} \)
41 \( 1 + (2.18 + 3.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 9.37T + 43T^{2} \)
47 \( 1 + 2T + 47T^{2} \)
53 \( 1 + (-5.74 + 9.94i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.55 - 7.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.05 + 3.56i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-5.74 - 9.94i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 2.62T + 73T^{2} \)
79 \( 1 + (0.686 + 1.18i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-4 + 6.92i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.81 + 3.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.74T + 97T^{2} \)
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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

−10.14337898426185327149967461275, −8.967264914079593855710299005238, −8.613284704411621195101645030205, −7.71227432957783973405319129947, −6.85991744377134526530857925141, −5.52745289105988487352002997138, −5.00353744265072309315259240100, −3.82281379231754222082983253769, −2.07124771388573396557134066804, −0.24314051823975801318320563392, 1.84796125687520303578304446809, 3.14534325015164058069313209400, 3.99698603976483273157925167795, 5.19824344144370949070611278026, 6.61238046984711373162120326806, 7.45861779093245605199772711702, 8.105140970445295276390709867040, 9.205889644758049796721342707368, 10.15870995350582251424255937715, 10.82827239006152848567741987123

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