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Properties

Label 2-666-9.4-c1-0-9
Degree $2$
Conductor $666$
Sign $-0.0351 - 0.999i$
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 + (−0.120 + 1.72i)3-s + (−0.499 + 0.866i)4-s + (0.181 − 0.314i)5-s + (1.55 − 0.759i)6-s + (2.51 + 4.36i)7-s + 0.999·8-s + (−2.97 − 0.416i)9-s − 0.363·10-s + (1.40 + 2.42i)11-s + (−1.43 − 0.968i)12-s + (1.98 − 3.44i)13-s + (2.51 − 4.36i)14-s + (0.521 + 0.351i)15-s + (−0.5 − 0.866i)16-s − 6.69·17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.0696 + 0.997i)3-s + (−0.249 + 0.433i)4-s + (0.0812 − 0.140i)5-s + (0.635 − 0.310i)6-s + (0.952 + 1.64i)7-s + 0.353·8-s + (−0.990 − 0.138i)9-s − 0.114·10-s + (0.422 + 0.731i)11-s + (−0.414 − 0.279i)12-s + (0.550 − 0.954i)13-s + (0.673 − 1.16i)14-s + (0.134 + 0.0908i)15-s + (−0.125 − 0.216i)16-s − 1.62·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0351 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0351 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.0351 - 0.999i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.0351 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.808328 + 0.837296i\)
\(L(\frac12)\) \(\approx\) \(0.808328 + 0.837296i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.120 - 1.72i)T \)
37 \( 1 + T \)
good5 \( 1 + (-0.181 + 0.314i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-2.51 - 4.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.40 - 2.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.98 + 3.44i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 6.69T + 17T^{2} \)
19 \( 1 - 4.01T + 19T^{2} \)
23 \( 1 + (2.61 - 4.53i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.09 + 5.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.38 - 4.12i)T + (-15.5 - 26.8i)T^{2} \)
41 \( 1 + (-0.969 + 1.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.533 + 0.924i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.40 - 4.16i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 8.18T + 53T^{2} \)
59 \( 1 + (3.74 - 6.48i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.597 - 1.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.35 - 4.08i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 0.867T + 71T^{2} \)
73 \( 1 + 7.28T + 73T^{2} \)
79 \( 1 + (-4.29 - 7.44i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.12 + 8.87i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 16.1T + 89T^{2} \)
97 \( 1 + (7.87 + 13.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.83247308134525509193031634237, −9.782985827477241361526784223506, −9.000177669269998759480488223914, −8.659546890945247594735483343210, −7.53752662430550010253085925531, −5.86323890536383235690623560178, −5.20744891745346969144047572186, −4.22539131992868549465825980338, −2.96732708966190618784736774459, −1.84706988655242206325499761994, 0.73329209591714173451359218478, 1.91720193321941790790997986872, 3.85338460690090373462779535440, 4.85722982604022905847331053230, 6.24820674677084733398294324246, 6.82944086904077452892662157402, 7.53549398852154690967042809015, 8.424070223600582542800241331233, 9.066528532878749892544227588851, 10.50906798213104942628100712883

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