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Properties

Label 2-1888-59.58-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9618243107\)
\(L(\frac12)\) \(\approx\) \(0.9618243107\)
\(L(1)\) 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

−8.935258016979554557086966210611, −8.329565441145867745267600018623, −8.007668359045917495199468547522, −6.97118716836826878741381088790, −6.08502691000805411565336766729, −5.18399997826802885615910152909, −3.83589018973651736621583961001, −3.18935511987486664868338396793, −2.78604466398144268302796112406, −0.61561566118903197714013457112, 1.82999944218405448201775953115, 2.92522619598164201816688738203, 3.74438954787452573061322633487, 4.36795708680920636858488903584, 5.57171166011373107475230065732, 6.83560628792965836756447953090, 7.27431198129954678663093090444, 7.999072951428401022286927639793, 8.912965140292611874739938664670, 9.567126484044643341493548491116

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