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Properties

Label 2-1904-17.16-c1-0-37
Degree $2$
Conductor $1904$
Sign $-0.00723 + 0.999i$
Analytic cond. $15.2035$
Root an. cond. $3.89916$
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  − 0.811i·3-s + 1.15i·5-s − i·7-s + 2.34·9-s − 2.12i·11-s − 1.15·13-s + 0.940·15-s + (−4.12 − 0.0298i)17-s − 0.653·19-s − 0.811·21-s − 7.65i·23-s + 3.65·25-s − 4.33i·27-s + 2.49i·29-s − 0.841i·31-s + ⋯
L(s)  = 1  − 0.468i·3-s + 0.518i·5-s − 0.377i·7-s + 0.780·9-s − 0.640i·11-s − 0.319·13-s + 0.242·15-s + (−0.999 − 0.00723i)17-s − 0.149·19-s − 0.177·21-s − 1.59i·23-s + 0.731·25-s − 0.834i·27-s + 0.464i·29-s − 0.151i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00723 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1904 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00723 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1904\)    =    \(2^{4} \cdot 7 \cdot 17\)
Sign: $-0.00723 + 0.999i$
Analytic conductor: \(15.2035\)
Root analytic conductor: \(3.89916\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1904} (1121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1904,\ (\ :1/2),\ -0.00723 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.521862189\)
\(L(\frac12)\) \(\approx\) \(1.521862189\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + iT \)
17 \( 1 + (4.12 + 0.0298i)T \)
good3 \( 1 + 0.811iT - 3T^{2} \)
5 \( 1 - 1.15iT - 5T^{2} \)
11 \( 1 + 2.12iT - 11T^{2} \)
13 \( 1 + 1.15T + 13T^{2} \)
19 \( 1 + 0.653T + 19T^{2} \)
23 \( 1 + 7.65iT - 23T^{2} \)
29 \( 1 - 2.49iT - 29T^{2} \)
31 \( 1 + 0.841iT - 31T^{2} \)
37 \( 1 - 2.97iT - 37T^{2} \)
41 \( 1 + 9.81iT - 41T^{2} \)
43 \( 1 - 7.03T + 43T^{2} \)
47 \( 1 + 6.55T + 47T^{2} \)
53 \( 1 - 6.90T + 53T^{2} \)
59 \( 1 - 0.929T + 59T^{2} \)
61 \( 1 + 5.49iT - 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 5.72iT - 71T^{2} \)
73 \( 1 + 6.99iT - 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 - 6.75T + 89T^{2} \)
97 \( 1 + 13.7iT - 97T^{2} \)
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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

−8.899737320229164044141923485203, −8.230126311017612458829335975409, −7.21583550487776805641742515376, −6.80288657951244653753031834070, −6.06801880085681866494573286086, −4.82235722110141348232554739261, −4.09056836443594198468091538016, −2.94522688704169676067134598848, −1.98094903059813534885106592864, −0.57937719610379861277392626493, 1.36117907392324372439476729160, 2.47104299714955391721565526555, 3.75778243075925309109626006152, 4.58793031850250260898848802360, 5.14133235122314314512913896769, 6.20892381922499571164497234907, 7.13329114320429119988440414981, 7.80001390025932578249146921597, 8.878684207989407503080546397306, 9.362879203307479637881809720663

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