Properties

Label 2-888-888.221-c0-0-5
Degree $2$
Conductor $888$
Sign $0.923 - 0.382i$
Analytic cond. $0.443169$
Root an. cond. $0.665709$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.382 + 0.923i)2-s − i·3-s + (−0.707 − 0.707i)4-s + 0.765i·5-s + (0.923 + 0.382i)6-s + 1.41·7-s + (0.923 − 0.382i)8-s − 9-s + (−0.707 − 0.292i)10-s + (−0.707 + 0.707i)12-s + (−0.541 + 1.30i)14-s + 0.765·15-s + i·16-s − 0.765·17-s + (0.382 − 0.923i)18-s + ⋯
L(s)  = 1  + (−0.382 + 0.923i)2-s − i·3-s + (−0.707 − 0.707i)4-s + 0.765i·5-s + (0.923 + 0.382i)6-s + 1.41·7-s + (0.923 − 0.382i)8-s − 9-s + (−0.707 − 0.292i)10-s + (−0.707 + 0.707i)12-s + (−0.541 + 1.30i)14-s + 0.765·15-s + i·16-s − 0.765·17-s + (0.382 − 0.923i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(888\)    =    \(2^{3} \cdot 3 \cdot 37\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(0.443169\)
Root analytic conductor: \(0.665709\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{888} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 888,\ (\ :0),\ 0.923 - 0.382i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8732717146\)
\(L(\frac12)\) \(\approx\) \(0.8732717146\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.382 - 0.923i)T \)
3 \( 1 + iT \)
37 \( 1 - iT \)
good5 \( 1 - 0.765iT - T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + 0.765T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.84T + T^{2} \)
29 \( 1 + 1.84iT - T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 - 1.84iT - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + 1.41iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.84T + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.49062950330606538512093187274, −9.180935791391954714200716688507, −8.444345513794112981302873800576, −7.74566231137189292363080506486, −7.05063091276249293108338472701, −6.36810167968014304933745054555, −5.35295341976006831416071683064, −4.44627615779020074799092500765, −2.66033044334883604330658577771, −1.34492471310294289283474034776, 1.38416934983643516095103266287, 2.78898341862503798062055291904, 4.01247690711548149581019082290, 4.92479305207624913519426102248, 5.20194692423636896820860405441, 7.13847645388817746087212401973, 8.298541574859129229382132422391, 8.819923983146561469502938961334, 9.298846245196592561679444907296, 10.50030654910265195606403652595

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