↑

Properties

Label 2-1452-33.26-c0-0-0
Degree $2$
Conductor $1452$
Sign $-0.511 - 0.859i$
Analytic cond. $0.724642$
Root an. cond. $0.851259$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.535 + 1.64i)7-s + (−0.809 − 0.587i)9-s + (−0.535 + 1.64i)19-s − 1.73·21-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−0.809 − 0.587i)31-s + (0.309 + 0.951i)37-s + (−1.61 + 1.17i)49-s + (−1.40 − 1.01i)57-s + (−1.40 + 1.01i)61-s + (0.535 − 1.64i)63-s + 67-s + (−0.535 − 1.64i)73-s + ⋯
L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.535 + 1.64i)7-s + (−0.809 − 0.587i)9-s + (−0.535 + 1.64i)19-s − 1.73·21-s + (0.309 − 0.951i)25-s + (0.809 − 0.587i)27-s + (−0.809 − 0.587i)31-s + (0.309 + 0.951i)37-s + (−1.61 + 1.17i)49-s + (−1.40 − 1.01i)57-s + (−1.40 + 1.01i)61-s + (0.535 − 1.64i)63-s + 67-s + (−0.535 − 1.64i)73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.511 - 0.859i$
Analytic conductor: \(0.724642\)
Root analytic conductor: \(0.851259\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :0),\ -0.511 - 0.859i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9236645258\)
\(L(\frac12)\) \(\approx\) \(0.9236645258\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (-0.309 + 0.951i)T^{2} \)
7 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (0.535 - 1.64i)T + (-0.809 - 0.587i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T + (0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 + 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (1.40 - 1.01i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T + T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.535 + 1.64i)T + (-0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.40 - 1.01i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
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

−9.955637168195141786645374302904, −9.180321858056496350366089227458, −8.516406705462750731364489937591, −7.891901861506978507500299044999, −6.32456365714121542176860089555, −5.82453243341425313453764768068, −5.03947306673242496577306360847, −4.16766357874219530227007174417, −3.05105618607425278216922040093, −1.97363974627272869150161852196, 0.797929629335933804442273657921, 1.97615291815694742462407394550, 3.32968426105099906696738074110, 4.48873775328346418570655791166, 5.24153292726985136725582093138, 6.43014598987565318990744650222, 7.17407400886441533313636799831, 7.52856537307586853781638586634, 8.505217621254905193552943499018, 9.356224825094436794364129320628

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