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Properties

Label 2-888-111.8-c1-0-12
Degree $2$
Conductor $888$
Sign $0.864 - 0.502i$
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  + (−1.31 − 1.12i)3-s + (1.00 + 3.73i)5-s + (1.76 − 3.05i)7-s + (0.453 + 2.96i)9-s − 4.23·11-s + (2.67 − 0.717i)13-s + (2.89 − 6.03i)15-s + (2.21 + 0.593i)17-s + (6.67 − 1.78i)19-s + (−5.75 + 2.02i)21-s + (−5.55 + 5.55i)23-s + (−8.60 + 4.96i)25-s + (2.75 − 4.40i)27-s + (4.01 + 4.01i)29-s + (−5.25 + 5.25i)31-s + ⋯
L(s)  = 1  + (−0.758 − 0.651i)3-s + (0.447 + 1.66i)5-s + (0.665 − 1.15i)7-s + (0.151 + 0.988i)9-s − 1.27·11-s + (0.742 − 0.198i)13-s + (0.748 − 1.55i)15-s + (0.537 + 0.143i)17-s + (1.53 − 0.410i)19-s + (−1.25 + 0.441i)21-s + (−1.15 + 1.15i)23-s + (−1.72 + 0.993i)25-s + (0.529 − 0.848i)27-s + (0.745 + 0.745i)29-s + (−0.944 + 0.944i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.502i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.864 - 0.502i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (785, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.864 - 0.502i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30084 + 0.350361i\)
\(L(\frac12)\) \(\approx\) \(1.30084 + 0.350361i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.31 + 1.12i)T \)
37 \( 1 + (-5.86 - 1.62i)T \)
good5 \( 1 + (-1.00 - 3.73i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.76 + 3.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 4.23T + 11T^{2} \)
13 \( 1 + (-2.67 + 0.717i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-2.21 - 0.593i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.67 + 1.78i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (5.55 - 5.55i)T - 23iT^{2} \)
29 \( 1 + (-4.01 - 4.01i)T + 29iT^{2} \)
31 \( 1 + (5.25 - 5.25i)T - 31iT^{2} \)
41 \( 1 + (-4.53 + 7.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.30 - 4.30i)T + 43iT^{2} \)
47 \( 1 - 5.68iT - 47T^{2} \)
53 \( 1 + (-2.63 + 1.52i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.14 - 2.18i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (1.77 + 6.63i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-13.0 - 7.50i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.60 - 1.50i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.53iT - 73T^{2} \)
79 \( 1 + (-3.18 + 0.852i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.0846 + 0.0488i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.48 + 9.28i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.73 + 6.73i)T + 97iT^{2} \)
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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

−10.42060044077875775480454750151, −9.818667564732349811880059872562, −8.004997035377593905072133250968, −7.47869292591583250234652011559, −6.95591554808393089002842748242, −5.85780255404795343297871017621, −5.23830942619752480142572187697, −3.70849766865442670258229219909, −2.61860530072363501328017308914, −1.25150637238173146361362550882, 0.842444415205125206740941387798, 2.31185291975213941895319974547, 4.04818110050165019528031678230, 5.00279375972750816139328086465, 5.53598262588231108583552985150, 6.02915656367174521225169900180, 7.898544719993328343951101342781, 8.410223582883337491378766557038, 9.382347044743132056734104407009, 9.840307454397779894888564501780

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