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Properties

Label 2-888-111.14-c1-0-32
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: $\chi_{888} (569, \cdot )$
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

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

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