Properties

Label 2-888-111.14-c1-0-21
Degree $2$
Conductor $888$
Sign $-0.124 + 0.992i$
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  + (−1.11 − 1.32i)3-s + (−0.892 + 3.33i)5-s + (−1.99 − 3.44i)7-s + (−0.532 + 2.95i)9-s + 3.36·11-s + (2.63 + 0.707i)13-s + (5.41 − 2.51i)15-s + (0.777 − 0.208i)17-s + (−7.12 − 1.90i)19-s + (−2.37 + 6.47i)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  + (−0.641 − 0.767i)3-s + (−0.399 + 1.49i)5-s + (−0.752 − 1.30i)7-s + (−0.177 + 0.984i)9-s + 1.01·11-s + (0.732 + 0.196i)13-s + (1.39 − 0.649i)15-s + (0.188 − 0.0505i)17-s + (−1.63 − 0.437i)19-s + (−0.517 + 1.41i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.124 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $-0.124 + 0.992i$
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: \((2,\ 888,\ (\ :1/2),\ -0.124 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.532288 - 0.603524i\)
\(L(\frac12)\) \(\approx\) \(0.532288 - 0.603524i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 1.32i)T \)
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

−10.29183462072222816330529507946, −9.103786560835584341245813285704, −7.81321491678816093900059673212, −7.13769355038310174652960005063, −6.53207273664786889216282372010, −6.07507038343364919356892323092, −4.23540002874403613556645130095, −3.57987483088951164280017721788, −2.18873327863605397445637270232, −0.46796252219486553791883354030, 1.26965011043543308198604822927, 3.21303380128067482703140201252, 4.25549774350996568277712105268, 4.99324096618023950009937167719, 6.00795148661014098293871378087, 6.47623903353924201175625254006, 8.264219390986621619575568536859, 8.947954216018037770111843044107, 9.198621391803855302244420637317, 10.30619573887790748958696240621

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