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Properties

Label 2-888-111.14-c1-0-16
Degree $2$
Conductor $888$
Sign $0.782 + 0.622i$
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  + (−1.68 − 0.390i)3-s + (0.623 − 2.32i)5-s + (0.882 + 1.52i)7-s + (2.69 + 1.31i)9-s + 6.17·11-s + (−2.69 − 0.721i)13-s + (−1.96 + 3.68i)15-s + (−4.97 + 1.33i)17-s + (0.561 + 0.150i)19-s + (−0.892 − 2.92i)21-s + (4.84 + 4.84i)23-s + (−0.700 − 0.404i)25-s + (−4.03 − 3.27i)27-s + (3.17 − 3.17i)29-s + (3.25 + 3.25i)31-s + ⋯
L(s)  = 1  + (−0.974 − 0.225i)3-s + (0.278 − 1.04i)5-s + (0.333 + 0.577i)7-s + (0.898 + 0.439i)9-s + 1.86·11-s + (−0.746 − 0.200i)13-s + (−0.506 + 0.951i)15-s + (−1.20 + 0.323i)17-s + (0.128 + 0.0344i)19-s + (−0.194 − 0.638i)21-s + (1.01 + 1.01i)23-s + (−0.140 − 0.0809i)25-s + (−0.776 − 0.630i)27-s + (0.589 − 0.589i)29-s + (0.585 + 0.585i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.782 + 0.622i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (569, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.782 + 0.622i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26651 - 0.442557i\)
\(L(\frac12)\) \(\approx\) \(1.26651 - 0.442557i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.390i)T \)
37 \( 1 + (-2.96 + 5.31i)T \)
good5 \( 1 + (-0.623 + 2.32i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.882 - 1.52i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 6.17T + 11T^{2} \)
13 \( 1 + (2.69 + 0.721i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.97 - 1.33i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.561 - 0.150i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.84 - 4.84i)T + 23iT^{2} \)
29 \( 1 + (-3.17 + 3.17i)T - 29iT^{2} \)
31 \( 1 + (-3.25 - 3.25i)T + 31iT^{2} \)
41 \( 1 + (1.88 + 3.26i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.31 + 1.31i)T - 43iT^{2} \)
47 \( 1 + 9.96iT - 47T^{2} \)
53 \( 1 + (-6.88 - 3.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-10.8 + 2.89i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.699 + 2.60i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.8 - 7.39i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-6.50 + 3.75i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.2iT - 73T^{2} \)
79 \( 1 + (9.05 + 2.42i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-0.219 - 0.126i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.58 - 5.92i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.9 - 10.9i)T - 97iT^{2} \)
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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

−9.979086252786161341869794557077, −9.093436771883116535300452815705, −8.637717435909045135899969328736, −7.26442349720770114029272132525, −6.53906507518606782859796953271, −5.57083181220659004898003902746, −4.87725570590938430151891630627, −4.00369295040381740966772661308, −2.03575304745865897383045473748, −0.965807037408797670559060074606, 1.13380127479055628378619922827, 2.73880081488136965336874323064, 4.18631545049650497990230548182, 4.70858682384353568084603434906, 6.16238036435291471258775986981, 6.78153768309713523460941475761, 7.15335377769451296941315718430, 8.715397582211913284352616544203, 9.613655626077830766766954953855, 10.30303656278012165985939283494

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