Properties

Label 2-888-111.14-c1-0-26
Degree $2$
Conductor $888$
Sign $0.561 + 0.827i$
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  + (1.10 − 1.33i)3-s + (0.721 − 2.69i)5-s + (2.47 + 4.29i)7-s + (−0.568 − 2.94i)9-s + 1.92·11-s + (1.17 + 0.315i)13-s + (−2.80 − 3.93i)15-s + (5.26 − 1.40i)17-s + (−4.56 − 1.22i)19-s + (8.46 + 1.42i)21-s + (2.41 + 2.41i)23-s + (−2.39 − 1.38i)25-s + (−4.56 − 2.48i)27-s + (−3.35 + 3.35i)29-s + (2.96 + 2.96i)31-s + ⋯
L(s)  = 1  + (0.636 − 0.771i)3-s + (0.322 − 1.20i)5-s + (0.936 + 1.62i)7-s + (−0.189 − 0.981i)9-s + 0.579·11-s + (0.326 + 0.0875i)13-s + (−0.723 − 1.01i)15-s + (1.27 − 0.341i)17-s + (−1.04 − 0.280i)19-s + (1.84 + 0.310i)21-s + (0.503 + 0.503i)23-s + (−0.479 − 0.276i)25-s + (−0.877 − 0.478i)27-s + (−0.622 + 0.622i)29-s + (0.532 + 0.532i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.561 + 0.827i$
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: \((2,\ 888,\ (\ :1/2),\ 0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.08243 - 1.10349i\)
\(L(\frac12)\) \(\approx\) \(2.08243 - 1.10349i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.10 + 1.33i)T \)
37 \( 1 + (5.71 + 2.08i)T \)
good5 \( 1 + (-0.721 + 2.69i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.47 - 4.29i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 1.92T + 11T^{2} \)
13 \( 1 + (-1.17 - 0.315i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-5.26 + 1.40i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.56 + 1.22i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.41 - 2.41i)T + 23iT^{2} \)
29 \( 1 + (3.35 - 3.35i)T - 29iT^{2} \)
31 \( 1 + (-2.96 - 2.96i)T + 31iT^{2} \)
41 \( 1 + (2.27 + 3.94i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.86 + 3.86i)T - 43iT^{2} \)
47 \( 1 - 1.47iT - 47T^{2} \)
53 \( 1 + (11.0 + 6.37i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (10.7 - 2.87i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.65 + 9.89i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-8.89 + 5.13i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.72 - 0.996i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.224iT - 73T^{2} \)
79 \( 1 + (-7.22 - 1.93i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (13.1 + 7.56i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-1.85 - 6.91i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (10.0 - 10.0i)T - 97iT^{2} \)
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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

−9.483609735040431070538454762948, −8.968176947909069414916907695009, −8.503362497743086013112999961613, −7.73225233006191502184301997508, −6.49748994919454208786515609855, −5.53973638949697993963172506485, −4.90910660125191072793460922455, −3.40995463257075069498132346600, −2.07750911748651283168697496482, −1.31703540876829914689488121709, 1.58518657446748467682068725363, 3.01351322159018175674508438297, 3.90837062674469143356675527572, 4.60577827253081850081424317151, 5.98617702650658491225521498676, 6.98704795472435945371048559938, 7.78686180341743580548944459048, 8.444414793975059760760931951901, 9.737491089181033738088003155651, 10.28817336153544107961442123700

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