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Properties

Label 2-888-111.14-c1-0-23
Degree $2$
Conductor $888$
Sign $0.679 + 0.733i$
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  + (0.567 − 1.63i)3-s + (0.0585 − 0.218i)5-s + (1.23 + 2.13i)7-s + (−2.35 − 1.85i)9-s + 4.45·11-s + (1.45 + 0.390i)13-s + (−0.324 − 0.219i)15-s + (−1.17 + 0.313i)17-s + (4.74 + 1.27i)19-s + (4.19 − 0.804i)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  + (0.327 − 0.944i)3-s + (0.0261 − 0.0977i)5-s + (0.465 + 0.806i)7-s + (−0.785 − 0.619i)9-s + 1.34·11-s + (0.404 + 0.108i)13-s + (−0.0837 − 0.0567i)15-s + (−0.284 + 0.0761i)17-s + (1.08 + 0.291i)19-s + (0.914 − 0.175i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.679 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.679 + 0.733i$
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: \((2,\ 888,\ (\ :1/2),\ 0.679 + 0.733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81708 - 0.793454i\)
\(L(\frac12)\) \(\approx\) \(1.81708 - 0.793454i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.567 + 1.63i)T \)
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

−9.694401765507585260270430180826, −9.095267736480799762356361091262, −8.288615344036026257881655552554, −7.61071297032190910696121669032, −6.41282268291141480671037610194, −6.01723749484223387885690352290, −4.68007881781829486039409137138, −3.44897586982168108805411487647, −2.24778421277366749089308782040, −1.16952822387330709675736605572, 1.35130531721762942282873499182, 3.06376783970799593294833195986, 3.93871937500163118343179721427, 4.71230013679596661123455492114, 5.75785049834338828420975447130, 6.92165121646553721579749637658, 7.72227991669779312310831957785, 8.906402725828375750756971722149, 9.206347885350936016739677024371, 10.37057673085047457807658824499

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