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Properties

Label 2-666-111.14-c1-0-7
Degree $2$
Conductor $666$
Sign $-0.358 + 0.933i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (−0.653 + 2.43i)5-s + (−1.76 − 3.05i)7-s + (−0.707 − 0.707i)8-s − 2.52·10-s − 2.53·11-s + (−2.84 − 0.762i)13-s + (2.49 − 2.49i)14-s + (0.500 − 0.866i)16-s + (1.73 − 0.463i)17-s + (−3.53 − 0.946i)19-s + (−0.653 − 2.43i)20-s + (−0.656 − 2.44i)22-s + (−3.15 − 3.15i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.292 + 1.09i)5-s + (−0.666 − 1.15i)7-s + (−0.249 − 0.249i)8-s − 0.798·10-s − 0.764·11-s + (−0.788 − 0.211i)13-s + (0.666 − 0.666i)14-s + (0.125 − 0.216i)16-s + (0.419 − 0.112i)17-s + (−0.809 − 0.217i)19-s + (−0.146 − 0.545i)20-s + (−0.139 − 0.521i)22-s + (−0.658 − 0.658i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.358 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $-0.358 + 0.933i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ -0.358 + 0.933i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0542814 - 0.0789563i\)
\(L(\frac12)\) \(\approx\) \(0.0542814 - 0.0789563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
37 \( 1 + (6.08 - 0.00578i)T \)
good5 \( 1 + (0.653 - 2.43i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.76 + 3.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 2.53T + 11T^{2} \)
13 \( 1 + (2.84 + 0.762i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.73 + 0.463i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (3.53 + 0.946i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.15 + 3.15i)T + 23iT^{2} \)
29 \( 1 + (0.0393 - 0.0393i)T - 29iT^{2} \)
31 \( 1 + (3.02 + 3.02i)T + 31iT^{2} \)
41 \( 1 + (-4.63 - 8.02i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.234 + 0.234i)T - 43iT^{2} \)
47 \( 1 + 4.39iT - 47T^{2} \)
53 \( 1 + (-3.09 - 1.78i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.18 - 2.46i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.55 + 9.52i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (13.5 - 7.80i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.56 - 2.05i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 4.09iT - 73T^{2} \)
79 \( 1 + (-2.84 - 0.762i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (7.86 + 4.54i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-3.78 - 14.1i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (0.841 - 0.841i)T - 97iT^{2} \)
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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

−10.38052756657648434473830580809, −9.506276679685842714451361288185, −8.162727912242060737507520843938, −7.39527366704529346084038574123, −6.86995946153389996326926008449, −5.96053706999823549901230430620, −4.68085289895640541727176438744, −3.67465967626637059888483013426, −2.68559317459148088107265259495, −0.04515174853428826234087614038, 1.90115294399461783874900591822, 3.03682928393788141046664146063, 4.30166319953181783098753243742, 5.27114982356675595507044390392, 5.93753181383144319698537450689, 7.43300867247320274712469319711, 8.515080629940045051635058000523, 9.070127722000197504055680085796, 9.896091021939731938422202659763, 10.76129785504004006157738200386

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