↑

Properties

Label 2-1850-37.36-c1-0-38
Degree $2$
Conductor $1850$
Sign $-0.674 + 0.737i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − i·2-s − 0.377·3-s − 4-s + 0.377i·6-s + 0.631·7-s + i·8-s − 2.85·9-s + 1.24·11-s + 0.377·12-s + 3.34i·13-s − 0.631i·14-s + 16-s − 3.10i·17-s + 2.85i·18-s + 5.97i·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.218·3-s − 0.5·4-s + 0.154i·6-s + 0.238·7-s + 0.353i·8-s − 0.952·9-s + 0.376·11-s + 0.109·12-s + 0.929i·13-s − 0.168i·14-s + 0.250·16-s − 0.753i·17-s + 0.673i·18-s + 1.37i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.674 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.674 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $-0.674 + 0.737i$
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: \((2,\ 1850,\ (\ :1/2),\ -0.674 + 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.022042710\)
\(L(\frac12)\) \(\approx\) \(1.022042710\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.48 - 4.10i)T \)
good3 \( 1 + 0.377T + 3T^{2} \)
7 \( 1 - 0.631T + 7T^{2} \)
11 \( 1 - 1.24T + 11T^{2} \)
13 \( 1 - 3.34iT - 13T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 - 5.97iT - 19T^{2} \)
23 \( 1 + 7.60iT - 23T^{2} \)
29 \( 1 + 9.57iT - 29T^{2} \)
31 \( 1 + 7.26iT - 31T^{2} \)
41 \( 1 + 8.45T + 41T^{2} \)
43 \( 1 + 4.86iT - 43T^{2} \)
47 \( 1 - 13.1T + 47T^{2} \)
53 \( 1 - 7.17T + 53T^{2} \)
59 \( 1 + 4.36iT - 59T^{2} \)
61 \( 1 - 2.14iT - 61T^{2} \)
67 \( 1 + 11.3T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 - 4.45T + 73T^{2} \)
79 \( 1 + 8.78iT - 79T^{2} \)
83 \( 1 - 6.63T + 83T^{2} \)
89 \( 1 + 13.7iT - 89T^{2} \)
97 \( 1 + 11.0iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.974249750791824416588693227000, −8.383598769247277035537627890198, −7.52367091080204843102778379366, −6.34209231668430686737010294519, −5.78251446768062499962619668527, −4.62692099417245303446780746687, −4.00321829539948268137440810155, −2.79877481040716668845787613707, −1.92947766902924657175391126621, −0.42823574282719631708011944008, 1.19934545260939534293543360697, 2.85751815096717008839936090787, 3.72996084344471194312992465840, 5.02674774445781648753245135357, 5.44473178453098803941772160582, 6.34383126742265129661700470706, 7.16005133053581942885480606058, 7.905771394668216002321395236494, 8.822569794070057753531382558759, 9.128286965004946876124430173948

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