Properties

Label 2-888-111.14-c1-0-14
Degree $2$
Conductor $888$
Sign $0.628 - 0.778i$
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  + (1.70 + 0.326i)3-s + (−0.0585 + 0.218i)5-s + (1.23 + 2.13i)7-s + (2.78 + 1.11i)9-s − 4.45·11-s + (1.45 + 0.390i)13-s + (−0.171 + 0.352i)15-s + (1.17 − 0.313i)17-s + (4.74 + 1.27i)19-s + (1.39 + 4.03i)21-s + (3.28 + 3.28i)23-s + (4.28 + 2.47i)25-s + (4.37 + 2.79i)27-s + (−5.98 + 5.98i)29-s + (−7.24 − 7.24i)31-s + ⋯
L(s)  = 1  + (0.982 + 0.188i)3-s + (−0.0261 + 0.0977i)5-s + (0.465 + 0.806i)7-s + (0.928 + 0.370i)9-s − 1.34·11-s + (0.404 + 0.108i)13-s + (−0.0441 + 0.0910i)15-s + (0.284 − 0.0761i)17-s + (1.08 + 0.291i)19-s + (0.305 + 0.879i)21-s + (0.685 + 0.685i)23-s + (0.857 + 0.494i)25-s + (0.842 + 0.538i)27-s + (−1.11 + 1.11i)29-s + (−1.30 − 1.30i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 - 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.628 - 0.778i$
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: \((2,\ 888,\ (\ :1/2),\ 0.628 - 0.778i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.03573 + 0.972838i\)
\(L(\frac12)\) \(\approx\) \(2.03573 + 0.972838i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 - 0.326i)T \)
37 \( 1 + (-5.17 - 3.20i)T \)
good5 \( 1 + (0.0585 - 0.218i)T + (-4.33 - 2.5i)T^{2} \)
7 \( 1 + (-1.23 - 2.13i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.45T + 11T^{2} \)
13 \( 1 + (-1.45 - 0.390i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (-1.17 + 0.313i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.74 - 1.27i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.28 - 3.28i)T + 23iT^{2} \)
29 \( 1 + (5.98 - 5.98i)T - 29iT^{2} \)
31 \( 1 + (7.24 + 7.24i)T + 31iT^{2} \)
41 \( 1 + (5.48 + 9.49i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.65 - 1.65i)T - 43iT^{2} \)
47 \( 1 - 1.96iT - 47T^{2} \)
53 \( 1 + (-9.27 - 5.35i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.54 + 1.75i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-3.26 + 12.1i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (12.3 - 7.14i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.52 - 2.03i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 8.68iT - 73T^{2} \)
79 \( 1 + (2.07 + 0.556i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (1.03 + 0.600i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (3.33 + 12.4i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-7.22 + 7.22i)T - 97iT^{2} \)
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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

−10.15555046918150209730803687017, −9.213847280559879712996976444919, −8.701394021855020492768262223183, −7.66734069635430886439045359909, −7.26772663857403833560913604639, −5.58274460061457966779611097185, −5.09550050427667041167904700224, −3.65551660008421958169794817665, −2.82467919881301202699257662269, −1.70836204120608289042482551136, 1.07271159599832361868288540558, 2.51119289073241425289857995091, 3.47806596603203977717261818087, 4.56023146421189754949597622295, 5.49776140632254067369475061265, 6.95304195218064718994802855255, 7.54379760760521514323948417419, 8.260620164223332763054378758016, 9.033687374930644310548343113687, 10.07180824363671599308014354278

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