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Properties

Label 2-666-111.80-c1-0-5
Degree $2$
Conductor $666$
Sign $0.588 - 0.808i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + 4.16·7-s + (−0.707 + 0.707i)8-s − 0.821·11-s + (0.418 + 0.418i)13-s + (2.94 + 2.94i)14-s − 1.00·16-s + (3.76 − 3.76i)17-s + (1.41 + 1.41i)19-s + (−0.581 − 0.581i)22-s + (−2.94 + 2.94i)23-s + 5i·25-s + 0.592i·26-s + 4.16i·28-s + (0.821 + 0.821i)29-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + 1.57·7-s + (−0.250 + 0.250i)8-s − 0.247·11-s + (0.116 + 0.116i)13-s + (0.786 + 0.786i)14-s − 0.250·16-s + (0.913 − 0.913i)17-s + (0.325 + 0.325i)19-s + (−0.123 − 0.123i)22-s + (−0.613 + 0.613i)23-s + i·25-s + 0.116i·26-s + 0.786i·28-s + (0.152 + 0.152i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.588 - 0.808i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (413, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01839 + 1.02764i\)
\(L(\frac12)\) \(\approx\) \(2.01839 + 1.02764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
37 \( 1 + (-5.58 + 2.41i)T \)
good5 \( 1 - 5iT^{2} \)
7 \( 1 - 4.16T + 7T^{2} \)
11 \( 1 + 0.821T + 11T^{2} \)
13 \( 1 + (-0.418 - 0.418i)T + 13iT^{2} \)
17 \( 1 + (-3.76 + 3.76i)T - 17iT^{2} \)
19 \( 1 + (-1.41 - 1.41i)T + 19iT^{2} \)
23 \( 1 + (2.94 - 2.94i)T - 23iT^{2} \)
29 \( 1 + (-0.821 - 0.821i)T + 29iT^{2} \)
31 \( 1 + (-2.16 + 2.16i)T - 31iT^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + (-0.837 - 0.837i)T + 43iT^{2} \)
47 \( 1 - 1.64iT - 47T^{2} \)
53 \( 1 - 8.35iT - 53T^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (4.32 - 4.32i)T - 61iT^{2} \)
67 \( 1 - 4.32iT - 67T^{2} \)
71 \( 1 + 7.53iT - 71T^{2} \)
73 \( 1 + 15.3iT - 73T^{2} \)
79 \( 1 + (7.48 + 7.48i)T + 79iT^{2} \)
83 \( 1 + 12.5iT - 83T^{2} \)
89 \( 1 + (8.00 + 8.00i)T + 89iT^{2} \)
97 \( 1 + (0.837 + 0.837i)T + 97iT^{2} \)
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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

−10.82204437216670399236292133242, −9.728383428384172807868337209003, −8.707542890485413277706099450263, −7.73697465972302214355260085994, −7.41309303803467983898146333558, −5.94143525034943546826027541638, −5.19349003617968536571557489490, −4.39077418574246677791199760030, −3.11450518233834422081898206454, −1.58449906497617445016768798068, 1.30374098371650650873677475797, 2.49385286914679194971768172749, 3.89338164091991299803473718401, 4.81927916152478493933909511787, 5.59320234819942129025053986287, 6.72089173446005419402329381498, 8.099736357025356292698345688603, 8.356195226419619838632131502390, 9.850435572334084790855802258063, 10.49815887404392094283012996184

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