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Properties

Label 2-666-111.80-c1-0-2
Degree $2$
Conductor $666$
Sign $0.894 - 0.447i$
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 − 2.16·7-s + (0.707 − 0.707i)8-s − 3.65·11-s + (3.58 + 3.58i)13-s + (1.52 + 1.52i)14-s − 1.00·16-s + (5.17 − 5.17i)17-s + (4.58 + 4.58i)19-s + (2.58 + 2.58i)22-s + (−1.52 + 1.52i)23-s + 5i·25-s − 5.06i·26-s − 2.16i·28-s + (3.65 + 3.65i)29-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s − 0.817·7-s + (0.250 − 0.250i)8-s − 1.10·11-s + (0.993 + 0.993i)13-s + (0.408 + 0.408i)14-s − 0.250·16-s + (1.25 − 1.25i)17-s + (1.05 + 1.05i)19-s + (0.550 + 0.550i)22-s + (−0.318 + 0.318i)23-s + i·25-s − 0.993i·26-s − 0.408i·28-s + (0.677 + 0.677i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.894 - 0.447i$
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: \((2,\ 666,\ (\ :1/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.933962 + 0.220543i\)
\(L(\frac12)\) \(\approx\) \(0.933962 + 0.220543i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.41 + 5.58i)T \)
good5 \( 1 - 5iT^{2} \)
7 \( 1 + 2.16T + 7T^{2} \)
11 \( 1 + 3.65T + 11T^{2} \)
13 \( 1 + (-3.58 - 3.58i)T + 13iT^{2} \)
17 \( 1 + (-5.17 + 5.17i)T - 17iT^{2} \)
19 \( 1 + (-4.58 - 4.58i)T + 19iT^{2} \)
23 \( 1 + (1.52 - 1.52i)T - 23iT^{2} \)
29 \( 1 + (-3.65 - 3.65i)T + 29iT^{2} \)
31 \( 1 + (4.16 - 4.16i)T - 31iT^{2} \)
41 \( 1 - 1.18T + 41T^{2} \)
43 \( 1 + (-7.16 - 7.16i)T + 43iT^{2} \)
47 \( 1 - 7.30iT - 47T^{2} \)
53 \( 1 - 14.0iT - 53T^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + (-8.32 + 8.32i)T - 61iT^{2} \)
67 \( 1 + 8.32iT - 67T^{2} \)
71 \( 1 + 10.3iT - 71T^{2} \)
73 \( 1 + 2.67iT - 73T^{2} \)
79 \( 1 + (-11.4 - 11.4i)T + 79iT^{2} \)
83 \( 1 + 9.76iT - 83T^{2} \)
89 \( 1 + (0.936 + 0.936i)T + 89iT^{2} \)
97 \( 1 + (7.16 + 7.16i)T + 97iT^{2} \)
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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

−10.58056746981089771439668343794, −9.527410196118007832307825136968, −9.268552489102958688466377704817, −7.894038033555811041112158474871, −7.35635198346205811037796600393, −6.11594665703030768459075514148, −5.13910414180964481110380474007, −3.64405380949980382551443213897, −2.90944228297862514351918876782, −1.27930907781113345460518625074, 0.69633307795331972571605770191, 2.65011755613759861859650421266, 3.79674846831425393873615543844, 5.36542099380426639544320421413, 5.94433479825075025479903964984, 6.96615012929977843246217218732, 8.045299381589839337006199674689, 8.431001553265448385312277340402, 9.789516448362855298192005248019, 10.18600260185061015500027104904

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