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Properties

Label 2-1888-59.58-c0-0-0
Degree $2$
Conductor $1888$
Sign $-i$
Analytic cond. $0.942234$
Root an. cond. $0.970687$
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  − 3-s − 5-s + 7-s + 1.41i·11-s − 1.41i·13-s + 15-s − 19-s − 21-s + 1.41i·23-s + 27-s + 29-s − 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·39-s + ⋯
L(s)  = 1  − 3-s − 5-s + 7-s + 1.41i·11-s − 1.41i·13-s + 15-s − 19-s − 21-s + 1.41i·23-s + 27-s + 29-s − 1.41i·33-s − 35-s + 1.41i·37-s + 1.41i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1888\)    =    \(2^{5} \cdot 59\)
Sign: $-i$
Analytic conductor: \(0.942234\)
Root analytic conductor: \(0.970687\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1888} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1888,\ (\ :0),\ -i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5111504678\)
\(L(\frac12)\) \(\approx\) \(0.5111504678\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
59 \( 1 + T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 + T + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 1.41iT - T^{2} \)
41 \( 1 + T + T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - T + T^{2} \)
61 \( 1 - 1.41iT - T^{2} \)
67 \( 1 - 1.41iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 - 1.41iT - T^{2} \)
89 \( 1 - 1.41iT - T^{2} \)
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

−9.852089591813853928835309259401, −8.550951089807183843708187606070, −7.979511113899550973424786685493, −7.35133015370066041643390713071, −6.42805889550096033045118583514, −5.40988162806161428764952057013, −4.83250821209533130616843293556, −4.07729212326524224375286879471, −2.80374303056836814814813590677, −1.33244986560295771247592084330, 0.46413613202372031524500103071, 2.08448191437344746400479906503, 3.53632811119413363569362669557, 4.45058467517575565253242398420, 5.03424764790364988943263660364, 6.16127650810643382583918606239, 6.62312195612586303083329015319, 7.71350078067324862489581751442, 8.598662510147989254445713855650, 8.772322615712581420711463191236

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