↑

Properties

Label 2-1850-5.4-c1-0-15
Degree $2$
Conductor $1850$
Sign $0.447 + 0.894i$
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 − 3.34i·3-s − 4-s − 3.34·6-s + 2.59i·7-s + i·8-s − 8.19·9-s + 4.74·11-s + 3.34i·12-s + 6.69i·13-s + 2.59·14-s + 16-s + 0.748i·17-s + 8.19i·18-s + 3.34·19-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.93i·3-s − 0.5·4-s − 1.36·6-s + 0.981i·7-s + 0.353i·8-s − 2.73·9-s + 1.43·11-s + 0.965i·12-s + 1.85i·13-s + 0.694·14-s + 0.250·16-s + 0.181i·17-s + 1.93i·18-s + 0.767·19-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1850\)    =    \(2 \cdot 5^{2} \cdot 37\)
Sign: $0.447 + 0.894i$
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: \((2,\ 1850,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.493063487\)
\(L(\frac12)\) \(\approx\) \(1.493063487\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.34iT - 3T^{2} \)
7 \( 1 - 2.59iT - 7T^{2} \)
11 \( 1 - 4.74T + 11T^{2} \)
13 \( 1 - 6.69iT - 13T^{2} \)
17 \( 1 - 0.748iT - 17T^{2} \)
19 \( 1 - 3.34T + 19T^{2} \)
23 \( 1 - 1.49iT - 23T^{2} \)
29 \( 1 + 3.94T + 29T^{2} \)
31 \( 1 - 7.79T + 31T^{2} \)
41 \( 1 + 6.44T + 41T^{2} \)
43 \( 1 - 1.94iT - 43T^{2} \)
47 \( 1 + 1.84iT - 47T^{2} \)
53 \( 1 - 10.4iT - 53T^{2} \)
59 \( 1 + 5.84T + 59T^{2} \)
61 \( 1 - 7.94T + 61T^{2} \)
67 \( 1 + 1.84iT - 67T^{2} \)
71 \( 1 + 3.88T + 71T^{2} \)
73 \( 1 + 7.49iT - 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 - 15.2iT - 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 - 10.4iT - 97T^{2} \)
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

−9.054856411104924971369612347189, −8.439522367075178374916625913177, −7.49797461339906262648856106222, −6.60729877887078619460438436599, −6.23219735339953370725548089926, −5.17570632084132790491013927666, −3.86692569781038100380642785662, −2.71823127919882464886203233547, −1.84800448447057828332884720395, −1.21967624179701784548455434461, 0.63390855109511972533099685245, 3.12547332349056135810887699352, 3.71267573026869637458518926580, 4.47668719851879990752589853102, 5.24765048083822574140661566258, 5.96039721595275308192581933611, 6.96505204834021500628011980233, 8.032148600391575467830185139138, 8.626824557963557544338016490205, 9.547224448363969298173710539116

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