↑

Properties

Label 2-44-44.35-c1-0-2
Degree $2$
Conductor $44$
Sign $0.0367 + 0.999i$
Analytic cond. $0.351341$
Root an. cond. $0.592740$
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  + (−1.40 + 0.124i)2-s + (−1.59 − 2.19i)3-s + (1.96 − 0.350i)4-s + (−0.720 − 2.21i)5-s + (2.52 + 2.90i)6-s + (1.04 + 0.758i)7-s + (−2.73 + 0.737i)8-s + (−1.35 + 4.17i)9-s + (1.29 + 3.03i)10-s + (3.29 + 0.387i)11-s + (−3.91 − 3.77i)12-s + (−0.279 − 0.0906i)13-s + (−1.56 − 0.939i)14-s + (−3.72 + 5.13i)15-s + (3.75 − 1.37i)16-s + (2.82 − 0.917i)17-s + ⋯
L(s)  = 1  + (−0.996 + 0.0878i)2-s + (−0.922 − 1.26i)3-s + (0.984 − 0.175i)4-s + (−0.322 − 0.992i)5-s + (1.03 + 1.18i)6-s + (0.394 + 0.286i)7-s + (−0.965 + 0.260i)8-s + (−0.452 + 1.39i)9-s + (0.408 + 0.960i)10-s + (0.993 + 0.116i)11-s + (−1.13 − 1.08i)12-s + (−0.0774 − 0.0251i)13-s + (−0.418 − 0.251i)14-s + (−0.962 + 1.32i)15-s + (0.938 − 0.344i)16-s + (0.684 − 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0367 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0367 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.0367 + 0.999i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1/2),\ 0.0367 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.309076 - 0.297910i\)
\(L(\frac12)\) \(\approx\) \(0.309076 - 0.297910i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.124i)T \)
11 \( 1 + (-3.29 - 0.387i)T \)
good3 \( 1 + (1.59 + 2.19i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.720 + 2.21i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.04 - 0.758i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (0.279 + 0.0906i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.82 + 0.917i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.38 + 1.00i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 0.525iT - 23T^{2} \)
29 \( 1 + (4.84 - 6.66i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.22 + 1.37i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (-4.22 - 3.07i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-3.28 - 4.52i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 3.49T + 43T^{2} \)
47 \( 1 + (-4.50 - 6.19i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.484 + 1.49i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.27 + 11.3i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (8.98 - 2.91i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 - 10.4iT - 67T^{2} \)
71 \( 1 + (3.41 - 1.11i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (-2.51 + 3.46i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.04 - 9.36i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.16 - 3.57i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 0.598T + 89T^{2} \)
97 \( 1 + (2.57 - 7.91i)T + (-78.4 - 57.0i)T^{2} \)
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

−16.32221652666639237633317651005, −14.65298571404682395365730336966, −12.82084520808606834123052305877, −11.98034823568191828408746098952, −11.23534988581062595934415152626, −9.314350753221014947975373387299, −8.042646164526603422197205714253, −6.90566390232419969627962759658, −5.48796192545704803665547124242, −1.30057736695860585162283973664, 3.76763754896625893689228736800, 5.96016753077809190510095146239, 7.44143757708028044726456244074, 9.277585551846816285134891311407, 10.36295616405696342024459286713, 11.13658093421463136069321872908, 11.92539755640892341438178282362, 14.54581583078982208534650299168, 15.31830415753538165920311104610, 16.52262113045919383427032922049

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