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Properties

Label 2-888-296.251-c1-0-21
Degree $2$
Conductor $888$
Sign $0.306 - 0.951i$
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 + i)2-s + (−0.866 − 0.5i)3-s − 2i·4-s + (0.5 − 0.133i)5-s + (1.36 − 0.366i)6-s + (2.36 + 1.36i)7-s + (2 + 2i)8-s + (0.499 + 0.866i)9-s + (−0.366 + 0.633i)10-s − 1.26i·11-s + (−1 + 1.73i)12-s + (−5.09 + 1.36i)13-s + (−3.73 + 1.00i)14-s + (−0.5 − 0.133i)15-s − 4·16-s + (5.96 + 1.59i)17-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)2-s + (−0.499 − 0.288i)3-s − i·4-s + (0.223 − 0.0599i)5-s + (0.557 − 0.149i)6-s + (0.894 + 0.516i)7-s + (0.707 + 0.707i)8-s + (0.166 + 0.288i)9-s + (−0.115 + 0.200i)10-s − 0.382i·11-s + (−0.288 + 0.499i)12-s + (−1.41 + 0.378i)13-s + (−0.997 + 0.267i)14-s + (−0.129 − 0.0345i)15-s − 16-s + (1.44 + 0.387i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.306 - 0.951i$
Analytic conductor: \(7.09071\)
Root analytic conductor: \(2.66283\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :1/2),\ 0.306 - 0.951i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.781034 + 0.568762i\)
\(L(\frac12)\) \(\approx\) \(0.781034 + 0.568762i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
37 \( 1 + (5.5 - 2.59i)T \)
good5 \( 1 + (-0.5 + 0.133i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-2.36 - 1.36i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 1.26iT - 11T^{2} \)
13 \( 1 + (5.09 - 1.36i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 + (-5.96 - 1.59i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-1.09 - 4.09i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.732 + 0.732i)T + 23iT^{2} \)
29 \( 1 + (-4.83 + 4.83i)T - 29iT^{2} \)
31 \( 1 + (-3.19 + 3.19i)T - 31iT^{2} \)
41 \( 1 + (-9.86 - 5.69i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + i)T - 43iT^{2} \)
47 \( 1 - 2.19iT - 47T^{2} \)
53 \( 1 + (11.1 - 6.46i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-13.6 - 3.66i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.13 - 7.96i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-11.3 - 6.56i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-10.0 - 5.83i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + (3.36 - 0.901i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-1.46 - 2.53i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.696 - 2.59i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (12.7 - 12.7i)T - 97iT^{2} \)
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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

−9.909369517543995986397512179305, −9.695295085970685707641061584299, −8.220995300823569612982492905187, −7.961335576221570987554692690456, −6.98176443771502595811018941147, −5.84883375878858118287122350232, −5.45158778738596685954402079941, −4.39423497552545598801057078373, −2.36804470058104458744359856534, −1.20258011970508642479085907656, 0.73836359122895138429544384410, 2.15229137424105223579925658080, 3.38060627064021495013109666427, 4.65368437821963455248805605088, 5.24236028948356383276878098289, 6.85338089396733834053705062811, 7.54241441450711061265993030038, 8.272146796916316202574478220073, 9.543268447283632471150469208624, 9.931474636563045276139988517322

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