Properties

Label 2-888-296.43-c1-0-15
Degree $2$
Conductor $888$
Sign $0.999 + 0.0258i$
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  + (−0.765 − 1.18i)2-s − i·3-s + (−0.829 + 1.81i)4-s + (−0.186 + 0.186i)5-s + (−1.18 + 0.765i)6-s − 1.09i·7-s + (2.79 − 0.405i)8-s − 9-s + (0.365 + 0.0792i)10-s + 3.62i·11-s + (1.81 + 0.829i)12-s + (−1.20 + 1.20i)13-s + (−1.30 + 0.841i)14-s + (0.186 + 0.186i)15-s + (−2.62 − 3.01i)16-s + (1.89 + 1.89i)17-s + ⋯
L(s)  = 1  + (−0.540 − 0.841i)2-s − 0.577i·3-s + (−0.414 + 0.909i)4-s + (−0.0835 + 0.0835i)5-s + (−0.485 + 0.312i)6-s − 0.415i·7-s + (0.989 − 0.143i)8-s − 0.333·9-s + (0.115 + 0.0250i)10-s + 1.09i·11-s + (0.525 + 0.239i)12-s + (−0.335 + 0.335i)13-s + (−0.349 + 0.224i)14-s + (0.0482 + 0.0482i)15-s + (−0.655 − 0.754i)16-s + (0.459 + 0.459i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.999 + 0.0258i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.999 + 0.0258i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919980 - 0.0119029i\)
\(L(\frac12)\) \(\approx\) \(0.919980 - 0.0119029i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.765 + 1.18i)T \)
3 \( 1 + iT \)
37 \( 1 + (-3.91 - 4.65i)T \)
good5 \( 1 + (0.186 - 0.186i)T - 5iT^{2} \)
7 \( 1 + 1.09iT - 7T^{2} \)
11 \( 1 - 3.62iT - 11T^{2} \)
13 \( 1 + (1.20 - 1.20i)T - 13iT^{2} \)
17 \( 1 + (-1.89 - 1.89i)T + 17iT^{2} \)
19 \( 1 + (-1.92 - 1.92i)T + 19iT^{2} \)
23 \( 1 + (2.87 - 2.87i)T - 23iT^{2} \)
29 \( 1 + (5.85 + 5.85i)T + 29iT^{2} \)
31 \( 1 + (-5.38 - 5.38i)T + 31iT^{2} \)
41 \( 1 + 8.54iT - 41T^{2} \)
43 \( 1 + (-4.19 - 4.19i)T + 43iT^{2} \)
47 \( 1 + 2.45iT - 47T^{2} \)
53 \( 1 - 7.38iT - 53T^{2} \)
59 \( 1 + (-4.17 - 4.17i)T + 59iT^{2} \)
61 \( 1 + (-3.57 - 3.57i)T + 61iT^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 - 3.40iT - 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 + (-9.37 + 9.37i)T - 79iT^{2} \)
83 \( 1 + 6.91T + 83T^{2} \)
89 \( 1 + (-3.29 + 3.29i)T - 89iT^{2} \)
97 \( 1 + (11.8 + 11.8i)T + 97iT^{2} \)
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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

−10.03531630175208992828713471147, −9.526703645330947491786625513071, −8.472351487413699491267436047194, −7.47777599211565033639086482894, −7.23681170107616966267257667658, −5.79021969299702612951109254522, −4.49786340733740648118700417501, −3.58223883887200204968342954414, −2.30653298266232893884344740707, −1.28496216068451211084242634937, 0.59210714438850883576409144473, 2.59319824521352599691666517972, 3.98465698576040402404851443119, 5.11264663530707576395613044269, 5.76029009288157561749614135816, 6.65201862523257523385623179184, 7.83594060976908252856169139699, 8.378972346542207978694745932180, 9.292547778706214997015093733903, 9.849450847073634806351684515826

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