↑

Properties

Label 2-1450-5.4-c1-0-22
Degree $2$
Conductor $1450$
Sign $0.894 + 0.447i$
Analytic cond. $11.5783$
Root an. cond. $3.40269$
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 + 3i·3-s − 4-s + 3·6-s − 2i·7-s + i·8-s − 6·9-s − 11-s − 3i·12-s − 3i·13-s − 2·14-s + 16-s − 4i·17-s + 6i·18-s + 8·19-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.73i·3-s − 0.5·4-s + 1.22·6-s − 0.755i·7-s + 0.353i·8-s − 2·9-s − 0.301·11-s − 0.866i·12-s − 0.832i·13-s − 0.534·14-s + 0.250·16-s − 0.970i·17-s + 1.41i·18-s + 1.83·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1450\)    =    \(2 \cdot 5^{2} \cdot 29\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(11.5783\)
Root analytic conductor: \(3.40269\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1450} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1450,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.406334211\)
\(L(\frac12)\) \(\approx\) \(1.406334211\)
\(L(\frac{3}{2})\) 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 \( 1 \)
29 \( 1 - T \)
good3 \( 1 - 3iT - 3T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + 3iT - 13T^{2} \)
17 \( 1 + 4iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
31 \( 1 - 3T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 + 7iT - 43T^{2} \)
47 \( 1 - 11iT - 47T^{2} \)
53 \( 1 + iT - 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 4T + 61T^{2} \)
67 \( 1 + 4iT - 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 - 12iT - 73T^{2} \)
79 \( 1 - 7T + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + 6iT - 97T^{2} \)
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

−9.656673978944485817945755109168, −9.105901273511124275896363865348, −8.070811854580155801325589679859, −7.22176613698114445616066508411, −5.61551160046940810347253820833, −5.11843124209932158720987072547, −4.24413816747651878412723367112, −3.42917031212217965257283703036, −2.74683218414730530315708067717, −0.68223598987335013418230784781, 1.12289180430134975203737351371, 2.21515529355177193071714087003, 3.34905528767074658152241303174, 4.92233860362208204119085756690, 5.77609085151204401797065639603, 6.43970947413454471191299971863, 7.11345316150251412397070581245, 7.891772921592812951214465013513, 8.451380509896245349717997037347, 9.227986290095227932557559102068

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