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Properties

Label 2-666-111.2-c1-0-4
Degree $2$
Conductor $666$
Sign $-0.310 - 0.950i$
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  + (−0.0871 + 0.996i)2-s + (−0.984 − 0.173i)4-s + (3.03 + 1.41i)5-s + (−1.63 + 0.593i)7-s + (0.258 − 0.965i)8-s + (−1.67 + 2.90i)10-s + (−0.408 − 0.708i)11-s + (3.82 + 5.46i)13-s + (−0.449 − 1.67i)14-s + (0.939 + 0.342i)16-s + (0.771 − 1.10i)17-s + (−1.63 + 0.143i)19-s + (−2.74 − 1.92i)20-s + (0.741 − 0.345i)22-s + (−4.85 + 1.30i)23-s + ⋯
L(s)  = 1  + (−0.0616 + 0.704i)2-s + (−0.492 − 0.0868i)4-s + (1.35 + 0.633i)5-s + (−0.616 + 0.224i)7-s + (0.0915 − 0.341i)8-s + (−0.529 + 0.917i)10-s + (−0.123 − 0.213i)11-s + (1.06 + 1.51i)13-s + (−0.120 − 0.447i)14-s + (0.234 + 0.0855i)16-s + (0.187 − 0.267i)17-s + (−0.375 + 0.0328i)19-s + (−0.613 − 0.429i)20-s + (0.158 − 0.0736i)22-s + (−1.01 + 0.271i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.310 - 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.937997 + 1.29250i\)
\(L(\frac12)\) \(\approx\) \(0.937997 + 1.29250i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0871 - 0.996i)T \)
3 \( 1 \)
37 \( 1 + (-4.15 + 4.44i)T \)
good5 \( 1 + (-3.03 - 1.41i)T + (3.21 + 3.83i)T^{2} \)
7 \( 1 + (1.63 - 0.593i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (0.408 + 0.708i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.82 - 5.46i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.771 + 1.10i)T + (-5.81 - 15.9i)T^{2} \)
19 \( 1 + (1.63 - 0.143i)T + (18.7 - 3.29i)T^{2} \)
23 \( 1 + (4.85 - 1.30i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-3.12 - 0.836i)T + (25.1 + 14.5i)T^{2} \)
31 \( 1 + (-5.52 - 5.52i)T + 31iT^{2} \)
41 \( 1 + (-0.631 + 3.58i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (7.38 - 7.38i)T - 43iT^{2} \)
47 \( 1 + (-2.69 - 1.55i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.64 + 4.52i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (3.21 + 6.88i)T + (-37.9 + 45.1i)T^{2} \)
61 \( 1 + (9.61 - 6.73i)T + (20.8 - 57.3i)T^{2} \)
67 \( 1 + (-1.99 - 5.49i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (-7.41 + 8.83i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 - 6.79iT - 73T^{2} \)
79 \( 1 + (-3.68 + 7.89i)T + (-50.7 - 60.5i)T^{2} \)
83 \( 1 + (8.80 - 1.55i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (-4.08 + 1.90i)T + (57.2 - 68.1i)T^{2} \)
97 \( 1 + (2.31 + 8.63i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.52116540288067922854407156702, −9.747830397555507408624298097260, −9.142016895996912735658251449463, −8.244586996864621068588364290256, −6.88136853938863919719383095917, −6.33082839685356611230835092693, −5.78010151470501071765184568256, −4.44496534228427732231999151505, −3.11045897145062963173916730866, −1.75659483971200939094235289221, 0.933503535513490081395738133199, 2.26213530565711702421744181374, 3.42283730446185456008271531629, 4.68531272595372787330756837558, 5.79929286630858198140352754515, 6.32213129362924190241202000256, 7.993457396300933595532357812675, 8.639205546785425437613326604960, 9.775425445840606929043134418239, 10.07844029233876562294351290274

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