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Properties

Label 2-888-111.8-c1-0-7
Degree $2$
Conductor $888$
Sign $-0.453 - 0.891i$
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  + (0.595 − 1.62i)3-s + (0.892 + 3.33i)5-s + (−1.99 + 3.44i)7-s + (−2.29 − 1.93i)9-s − 3.36·11-s + (2.63 − 0.707i)13-s + (5.95 + 0.532i)15-s + (−0.777 − 0.208i)17-s + (−7.12 + 1.90i)19-s + (4.42 + 5.29i)21-s + (−1.05 + 1.05i)23-s + (−5.97 + 3.44i)25-s + (−4.51 + 2.57i)27-s + (−4.97 − 4.97i)29-s + (−6.17 + 6.17i)31-s + ⋯
L(s)  = 1  + (0.343 − 0.939i)3-s + (0.399 + 1.49i)5-s + (−0.752 + 1.30i)7-s + (−0.763 − 0.645i)9-s − 1.01·11-s + (0.732 − 0.196i)13-s + (1.53 + 0.137i)15-s + (−0.188 − 0.0505i)17-s + (−1.63 + 0.437i)19-s + (0.965 + 1.15i)21-s + (−0.219 + 0.219i)23-s + (−1.19 + 0.689i)25-s + (−0.868 + 0.495i)27-s + (−0.924 − 0.924i)29-s + (−1.10 + 1.10i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.453 - 0.891i$
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: \((2,\ 888,\ (\ :1/2),\ -0.453 - 0.891i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.498485 + 0.812791i\)
\(L(\frac12)\) \(\approx\) \(0.498485 + 0.812791i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.595 + 1.62i)T \)
37 \( 1 + (-4.71 - 3.84i)T \)
good5 \( 1 + (-0.892 - 3.33i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (1.99 - 3.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + 3.36T + 11T^{2} \)
13 \( 1 + (-2.63 + 0.707i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (0.777 + 0.208i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (7.12 - 1.90i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.05 - 1.05i)T - 23iT^{2} \)
29 \( 1 + (4.97 + 4.97i)T + 29iT^{2} \)
31 \( 1 + (6.17 - 6.17i)T - 31iT^{2} \)
41 \( 1 + (-4.81 + 8.33i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-7.91 - 7.91i)T + 43iT^{2} \)
47 \( 1 - 1.70iT - 47T^{2} \)
53 \( 1 + (-3.10 + 1.79i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.07 - 1.89i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.03 - 11.3i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (7.22 + 4.17i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.70 - 0.984i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.15iT - 73T^{2} \)
79 \( 1 + (-16.7 + 4.48i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-0.914 + 0.528i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.276 + 1.03i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.31 - 2.31i)T + 97iT^{2} \)
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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

−10.52892953681492726293904611535, −9.471219798823774007168602125539, −8.675484205413256375293262078569, −7.79709768288814673984581050612, −6.90545104874057507734035033177, −6.02516537636469381407739943816, −5.78630823543502356515687601633, −3.66164479936688486073889523908, −2.63599924020474522385881703731, −2.19820410945322013298089768331, 0.39818856395874185648241105765, 2.20691502196767700042843155717, 3.78806705706768027266518606905, 4.31733262856620504101530436061, 5.28535211737131989321168145336, 6.18389549618227410757941496776, 7.51884648007392152030901347674, 8.397400341958282925011786941525, 9.139365843636671473044745706294, 9.702086767902986232511941673330

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