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Properties

Label 2-666-111.35-c1-0-10
Degree $2$
Conductor $666$
Sign $0.931 + 0.364i$
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  + (0.996 + 0.0871i)2-s + (0.984 + 0.173i)4-s + (0.567 − 1.21i)5-s + (−0.300 + 0.109i)7-s + (0.965 + 0.258i)8-s + (0.671 − 1.16i)10-s + (−0.282 − 0.489i)11-s + (4.64 − 3.25i)13-s + (−0.309 + 0.0828i)14-s + (0.939 + 0.342i)16-s + (2.15 + 1.50i)17-s + (0.251 + 2.87i)19-s + (0.770 − 1.10i)20-s + (−0.238 − 0.512i)22-s + (−2.16 − 8.09i)23-s + ⋯
L(s)  = 1  + (0.704 + 0.0616i)2-s + (0.492 + 0.0868i)4-s + (0.253 − 0.544i)5-s + (−0.113 + 0.0413i)7-s + (0.341 + 0.0915i)8-s + (0.212 − 0.367i)10-s + (−0.0852 − 0.147i)11-s + (1.28 − 0.901i)13-s + (−0.0826 + 0.0221i)14-s + (0.234 + 0.0855i)16-s + (0.522 + 0.365i)17-s + (0.0577 + 0.660i)19-s + (0.172 − 0.246i)20-s + (−0.0509 − 0.109i)22-s + (−0.452 − 1.68i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 + 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(666\)    =    \(2 \cdot 3^{2} \cdot 37\)
Sign: $0.931 + 0.364i$
Analytic conductor: \(5.31803\)
Root analytic conductor: \(2.30608\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{666} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 666,\ (\ :1/2),\ 0.931 + 0.364i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42646 - 0.458178i\)
\(L(\frac12)\) \(\approx\) \(2.42646 - 0.458178i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.996 - 0.0871i)T \)
3 \( 1 \)
37 \( 1 + (-1.50 - 5.89i)T \)
good5 \( 1 + (-0.567 + 1.21i)T + (-3.21 - 3.83i)T^{2} \)
7 \( 1 + (0.300 - 0.109i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (0.282 + 0.489i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.64 + 3.25i)T + (4.44 - 12.2i)T^{2} \)
17 \( 1 + (-2.15 - 1.50i)T + (5.81 + 15.9i)T^{2} \)
19 \( 1 + (-0.251 - 2.87i)T + (-18.7 + 3.29i)T^{2} \)
23 \( 1 + (2.16 + 8.09i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + (-0.626 + 2.33i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (-1.25 + 1.25i)T - 31iT^{2} \)
41 \( 1 + (0.936 - 5.31i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + (-0.271 - 0.271i)T + 43iT^{2} \)
47 \( 1 + (8.86 + 5.11i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.41 + 3.88i)T + (-40.6 - 34.0i)T^{2} \)
59 \( 1 + (2.87 - 1.33i)T + (37.9 - 45.1i)T^{2} \)
61 \( 1 + (1.12 + 1.60i)T + (-20.8 + 57.3i)T^{2} \)
67 \( 1 + (-3.00 - 8.25i)T + (-51.3 + 43.0i)T^{2} \)
71 \( 1 + (2.58 - 3.07i)T + (-12.3 - 69.9i)T^{2} \)
73 \( 1 - 2.39iT - 73T^{2} \)
79 \( 1 + (-8.72 - 4.06i)T + (50.7 + 60.5i)T^{2} \)
83 \( 1 + (10.5 - 1.86i)T + (77.9 - 28.3i)T^{2} \)
89 \( 1 + (0.359 + 0.770i)T + (-57.2 + 68.1i)T^{2} \)
97 \( 1 + (-0.0722 + 0.0193i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.52047271322019846343268978105, −9.771587455521808031605788853226, −8.411284928757900667642799888834, −8.088696266282488857140524119833, −6.57743709357873606390214822318, −5.92727111310484773853826852058, −5.02555776245571634523562803483, −3.92620995406557434414447629268, −2.89717372713984788306633465792, −1.28905679012062702302207769867, 1.64822192114859147777216712504, 3.02584191726295026771888497327, 3.92561148713508175727498052412, 5.10445508072905059486892566692, 6.10841444719560752251027482638, 6.82061035027149955775921597368, 7.74077619226858295183345909040, 8.953240483007946174786246980569, 9.792508900701941735524229717151, 10.80142361365102252256124583013

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