Properties

Label 2-888-111.14-c1-0-19
Degree $2$
Conductor $888$
Sign $0.986 - 0.162i$
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  + (0.169 + 1.72i)3-s + (0.336 − 1.25i)5-s + (−1.20 − 2.09i)7-s + (−2.94 + 0.584i)9-s − 0.238·11-s + (0.935 + 0.250i)13-s + (2.21 + 0.366i)15-s + (6.67 − 1.78i)17-s + (3.10 + 0.831i)19-s + (3.40 − 2.43i)21-s + (1.92 + 1.92i)23-s + (2.86 + 1.65i)25-s + (−1.50 − 4.97i)27-s + (3.98 − 3.98i)29-s + (5.62 + 5.62i)31-s + ⋯
L(s)  = 1  + (0.0979 + 0.995i)3-s + (0.150 − 0.560i)5-s + (−0.456 − 0.791i)7-s + (−0.980 + 0.194i)9-s − 0.0720·11-s + (0.259 + 0.0694i)13-s + (0.572 + 0.0946i)15-s + (1.61 − 0.433i)17-s + (0.711 + 0.190i)19-s + (0.742 − 0.532i)21-s + (0.400 + 0.400i)23-s + (0.573 + 0.331i)25-s + (−0.290 − 0.957i)27-s + (0.739 − 0.739i)29-s + (1.01 + 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.162i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.986 - 0.162i$
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: \((2,\ 888,\ (\ :1/2),\ 0.986 - 0.162i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61711 + 0.132087i\)
\(L(\frac12)\) \(\approx\) \(1.61711 + 0.132087i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.169 - 1.72i)T \)
37 \( 1 + (-6.08 + 0.00225i)T \)
good5 \( 1 + (-0.336 + 1.25i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (1.20 + 2.09i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.238T + 11T^{2} \)
13 \( 1 + (-0.935 - 0.250i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-6.67 + 1.78i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-3.10 - 0.831i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-1.92 - 1.92i)T + 23iT^{2} \)
29 \( 1 + (-3.98 + 3.98i)T - 29iT^{2} \)
31 \( 1 + (-5.62 - 5.62i)T + 31iT^{2} \)
41 \( 1 + (5.86 + 10.1i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.21 - 5.21i)T - 43iT^{2} \)
47 \( 1 + 7.53iT - 47T^{2} \)
53 \( 1 + (-0.441 - 0.255i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (11.0 - 2.97i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.30 + 4.86i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.58 - 2.06i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-5.78 + 3.34i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 10.8iT - 73T^{2} \)
79 \( 1 + (-12.1 - 3.24i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-4.33 - 2.50i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.463 + 1.73i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (4.84 - 4.84i)T - 97iT^{2} \)
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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

−10.00539633125420210737699919288, −9.552358900996036175508090886551, −8.578114993062098286000151773172, −7.77007687375035172222916084478, −6.68464955243303861195886252177, −5.49902921533673389656915735297, −4.88115732533078474246402675068, −3.74220280560944671988828461898, −3.00775764410956536957848987713, −0.990461472630180708966516033110, 1.17774502318434249452792488908, 2.69546053879554845279618559703, 3.22654238682453930108503861863, 5.02002535890453326657920017108, 6.08766381544661429402078746863, 6.49908904934483512805603377877, 7.62751183031242868161807393416, 8.251000561871852895049337169585, 9.223592855712384731239138566478, 10.06438469048857745460686108824

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