Properties

Label 2-1850-37.36-c1-0-26
Degree $2$
Conductor $1850$
Sign $0.180 - 0.983i$
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 − 1.78·3-s − 4-s − 1.78i·6-s + 3.14·7-s i·8-s + 0.191·9-s − 0.908·11-s + 1.78·12-s + 2.22i·13-s + 3.14i·14-s + 16-s − 2.10i·17-s + 0.191i·18-s + 4.16i·19-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.03·3-s − 0.5·4-s − 0.729i·6-s + 1.19·7-s − 0.353i·8-s + 0.0638·9-s − 0.274·11-s + 0.515·12-s + 0.616i·13-s + 0.841i·14-s + 0.250·16-s − 0.509i·17-s + 0.0451i·18-s + 0.955i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.180 - 0.983i$
Analytic conductor: \(14.7723\)
Root analytic conductor: \(3.84347\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1850} (1701, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1850,\ (\ :1/2),\ 0.180 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.117988875\)
\(L(\frac12)\) \(\approx\) \(1.117988875\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.98 - 1.10i)T \)
good3 \( 1 + 1.78T + 3T^{2} \)
7 \( 1 - 3.14T + 7T^{2} \)
11 \( 1 + 0.908T + 11T^{2} \)
13 \( 1 - 2.22iT - 13T^{2} \)
17 \( 1 + 2.10iT - 17T^{2} \)
19 \( 1 - 4.16iT - 19T^{2} \)
23 \( 1 + 7.66iT - 23T^{2} \)
29 \( 1 + 2.69iT - 29T^{2} \)
31 \( 1 - 5.96iT - 31T^{2} \)
41 \( 1 - 2.32T + 41T^{2} \)
43 \( 1 + 5.72iT - 43T^{2} \)
47 \( 1 + 8.89T + 47T^{2} \)
53 \( 1 - 9.37T + 53T^{2} \)
59 \( 1 - 5.55iT - 59T^{2} \)
61 \( 1 - 3.16iT - 61T^{2} \)
67 \( 1 - 7.64T + 67T^{2} \)
71 \( 1 - 13.5T + 71T^{2} \)
73 \( 1 - 2.14T + 73T^{2} \)
79 \( 1 - 3.35iT - 79T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 - 8.35iT - 89T^{2} \)
97 \( 1 - 5.14iT - 97T^{2} \)
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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

−9.287783232556759325353721453066, −8.379004217246320234768253459219, −7.920077340015483498500763331940, −6.83683016709868655477989127913, −6.27906862407865928768360790523, −5.29562589216130116489390772614, −4.88514975899101439279762439727, −3.97352578042919934351933256128, −2.35952437458870345060766743344, −0.902365098451485062900719993233, 0.66195650389792710045900698589, 1.83496395446087431319784346412, 3.02670033195858334089310797697, 4.21548925724925889436316796775, 5.12188776145105156784129258389, 5.51155573287086273415316971564, 6.52763057364499427098107733826, 7.72159254639401434841674236560, 8.203936383178404760615196828422, 9.241982515165820151674755528352

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