Properties

Label 2-1850-5.4-c1-0-24
Degree $2$
Conductor $1850$
Sign $0.894 + 0.447i$
Analytic cond. $14.7723$
Root an. cond. $3.84347$
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  i·2-s + 0.140i·3-s − 4-s + 0.140·6-s − 1.14i·7-s + i·8-s + 2.98·9-s + 3.12·11-s − 0.140i·12-s + 2.85i·13-s − 1.14·14-s + 16-s + 2.14i·17-s − 2.98i·18-s − 4.26·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.0810i·3-s − 0.5·4-s + 0.0573·6-s − 0.431i·7-s + 0.353i·8-s + 0.993·9-s + 0.940·11-s − 0.0405i·12-s + 0.793i·13-s − 0.304·14-s + 0.250·16-s + 0.519i·17-s − 0.702i·18-s − 0.977·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1850} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.844514358\)
\(L(\frac12)\) \(\approx\) \(1.844514358\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
5 \( 1 \)
37 \( 1 + iT \)
good3 \( 1 - 0.140iT - 3T^{2} \)
7 \( 1 + 1.14iT - 7T^{2} \)
11 \( 1 - 3.12T + 11T^{2} \)
13 \( 1 - 2.85iT - 13T^{2} \)
17 \( 1 - 2.14iT - 17T^{2} \)
19 \( 1 + 4.26T + 19T^{2} \)
23 \( 1 - 1.71iT - 23T^{2} \)
29 \( 1 - 4.28T + 29T^{2} \)
31 \( 1 - 6.40T + 31T^{2} \)
41 \( 1 + 3.24T + 41T^{2} \)
43 \( 1 - 10.6iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 7.96iT - 53T^{2} \)
59 \( 1 + 2.69T + 59T^{2} \)
61 \( 1 - 4.57T + 61T^{2} \)
67 \( 1 + 10.3iT - 67T^{2} \)
71 \( 1 + 7.10T + 71T^{2} \)
73 \( 1 + 0.261iT - 73T^{2} \)
79 \( 1 - 8.52T + 79T^{2} \)
83 \( 1 + 1.16iT - 83T^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + 14.2iT - 97T^{2} \)
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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

−9.316965519631754450519079713227, −8.606212556013625281223511902391, −7.67534415568493889994200881285, −6.72801673407453375527505570747, −6.13953829861419157266779605859, −4.57831908574141335514919526129, −4.31934063094934198991826669281, −3.34621901241882838187594418337, −1.99738701482508683225535501905, −1.10989498125410361064204116807, 0.871123833929993193664677498304, 2.29858666298445224730728806971, 3.62740916903624920960785403376, 4.48920825732716379166754598802, 5.28457720080031141584412424553, 6.35380490687197012137734532827, 6.78491385085090804871224464730, 7.68942290744558939679998667059, 8.520124598581952190023044170858, 9.078183855636291450777390601682

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