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Properties

Label 2-888-37.12-c1-0-17
Degree $2$
Conductor $888$
Sign $0.805 + 0.592i$
Analytic cond. $7.09071$
Root an. cond. $2.66283$
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  + (0.766 + 0.642i)3-s + (3.41 − 1.24i)5-s + (−2.06 + 0.752i)7-s + (0.173 + 0.984i)9-s + (−2.44 − 4.23i)11-s + (1.12 − 6.35i)13-s + (3.41 + 1.24i)15-s + (−1.15 − 6.55i)17-s + (6.22 + 5.22i)19-s + (−2.06 − 0.752i)21-s + (1.09 − 1.90i)23-s + (6.27 − 5.26i)25-s + (−0.500 + 0.866i)27-s + (3.05 + 5.29i)29-s − 1.06·31-s + ⋯
L(s)  = 1  + (0.442 + 0.371i)3-s + (1.52 − 0.555i)5-s + (−0.781 + 0.284i)7-s + (0.0578 + 0.328i)9-s + (−0.736 − 1.27i)11-s + (0.310 − 1.76i)13-s + (0.881 + 0.320i)15-s + (−0.280 − 1.58i)17-s + (1.42 + 1.19i)19-s + (−0.451 − 0.164i)21-s + (0.229 − 0.397i)23-s + (1.25 − 1.05i)25-s + (−0.0962 + 0.166i)27-s + (0.567 + 0.982i)29-s − 0.191·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.805 + 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.805 + 0.592i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.805 + 0.592i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01917 - 0.662871i\)
\(L(\frac12)\) \(\approx\) \(2.01917 - 0.662871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (0.272 - 6.07i)T \)
good5 \( 1 + (-3.41 + 1.24i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (2.06 - 0.752i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.44 + 4.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.12 + 6.35i)T + (-12.2 - 4.44i)T^{2} \)
17 \( 1 + (1.15 + 6.55i)T + (-15.9 + 5.81i)T^{2} \)
19 \( 1 + (-6.22 - 5.22i)T + (3.29 + 18.7i)T^{2} \)
23 \( 1 + (-1.09 + 1.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.05 - 5.29i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.06T + 31T^{2} \)
41 \( 1 + (0.391 - 2.21i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 + 2.91T + 43T^{2} \)
47 \( 1 + (1.92 - 3.34i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.976 - 0.355i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (-9.96 - 3.62i)T + (45.1 + 37.9i)T^{2} \)
61 \( 1 + (1.27 - 7.22i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (2.25 - 0.822i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (-0.202 - 0.169i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 3.97T + 73T^{2} \)
79 \( 1 + (1.63 - 0.595i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.645 + 3.66i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-12.1 - 4.40i)T + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (-4.45 + 7.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.09884612953323520761291773915, −9.245213099781676515134910756598, −8.581124706095340747830138237977, −7.69233780196789199793272902982, −6.32684022878006127748982466039, −5.48081341711199954874053557735, −5.14833274322065370470732510576, −3.13183224549520465511665949097, −2.86248445779061680038404475077, −1.02124441916503383381022153786, 1.76097704305436014179821258206, 2.42310497270677450465047135958, 3.70653263146332391904460114161, 4.97921150906213996483223353610, 6.15624851638853069091361528542, 6.76877082108560141548270701996, 7.38234737338064862930852234236, 8.764533498454182078148367209100, 9.601639466900312054918811579044, 9.889926027702811708815435812347

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