↑

Properties

Label 2-44-11.7-c2-0-1
Degree $2$
Conductor $44$
Sign $0.822 + 0.569i$
Analytic cond. $1.19891$
Root an. cond. $1.09494$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.680 − 2.09i)3-s + (3.38 − 2.45i)5-s + (−1.05 + 0.342i)7-s + (3.36 + 2.44i)9-s + (−10.1 + 4.34i)11-s + (−7.28 + 10.0i)13-s + (−2.84 − 8.74i)15-s + (6.97 + 9.59i)17-s + (−25.1 − 8.17i)19-s + 2.43i·21-s + 19.9·23-s + (−2.32 + 7.16i)25-s + (23.4 − 17.0i)27-s + (46.0 − 14.9i)29-s + (−13.9 − 10.1i)31-s + ⋯
L(s)  = 1  + (0.226 − 0.697i)3-s + (0.676 − 0.491i)5-s + (−0.150 + 0.0489i)7-s + (0.373 + 0.271i)9-s + (−0.918 + 0.395i)11-s + (−0.560 + 0.771i)13-s + (−0.189 − 0.583i)15-s + (0.410 + 0.564i)17-s + (−1.32 − 0.430i)19-s + 0.116i·21-s + 0.867·23-s + (−0.0931 + 0.286i)25-s + (0.867 − 0.630i)27-s + (1.58 − 0.516i)29-s + (−0.451 − 0.327i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(1.19891\)
Root analytic conductor: \(1.09494\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1),\ 0.822 + 0.569i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.15887 - 0.362135i\)
\(L(\frac12)\) \(\approx\) \(1.15887 - 0.362135i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (10.1 - 4.34i)T \)
good3 \( 1 + (-0.680 + 2.09i)T + (-7.28 - 5.29i)T^{2} \)
5 \( 1 + (-3.38 + 2.45i)T + (7.72 - 23.7i)T^{2} \)
7 \( 1 + (1.05 - 0.342i)T + (39.6 - 28.8i)T^{2} \)
13 \( 1 + (7.28 - 10.0i)T + (-52.2 - 160. i)T^{2} \)
17 \( 1 + (-6.97 - 9.59i)T + (-89.3 + 274. i)T^{2} \)
19 \( 1 + (25.1 + 8.17i)T + (292. + 212. i)T^{2} \)
23 \( 1 - 19.9T + 529T^{2} \)
29 \( 1 + (-46.0 + 14.9i)T + (680. - 494. i)T^{2} \)
31 \( 1 + (13.9 + 10.1i)T + (296. + 913. i)T^{2} \)
37 \( 1 + (22.4 + 69.0i)T + (-1.10e3 + 804. i)T^{2} \)
41 \( 1 + (39.2 + 12.7i)T + (1.35e3 + 988. i)T^{2} \)
43 \( 1 - 57.9iT - 1.84e3T^{2} \)
47 \( 1 + (2.70 - 8.32i)T + (-1.78e3 - 1.29e3i)T^{2} \)
53 \( 1 + (0.209 + 0.152i)T + (868. + 2.67e3i)T^{2} \)
59 \( 1 + (13.4 + 41.4i)T + (-2.81e3 + 2.04e3i)T^{2} \)
61 \( 1 + (48.1 + 66.2i)T + (-1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 - 52.4T + 4.48e3T^{2} \)
71 \( 1 + (-53.6 + 38.9i)T + (1.55e3 - 4.79e3i)T^{2} \)
73 \( 1 + (-55.0 + 17.8i)T + (4.31e3 - 3.13e3i)T^{2} \)
79 \( 1 + (-5.05 + 6.95i)T + (-1.92e3 - 5.93e3i)T^{2} \)
83 \( 1 + (-51.3 - 70.6i)T + (-2.12e3 + 6.55e3i)T^{2} \)
89 \( 1 + 3.65T + 7.92e3T^{2} \)
97 \( 1 + (5.54 + 4.03i)T + (2.90e3 + 8.94e3i)T^{2} \)
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

−15.55935110654442119762894501247, −14.20012015137154565356569652575, −13.05410491429986393948749297654, −12.51960448510881926999095605206, −10.66848372096428936630701192599, −9.398612861051052158177815056939, −7.973479365169510462808164078594, −6.61702567122488069200791151544, −4.86533072325521423805011366379, −2.09109604536586864639550248644, 3.02573544564853285017874901628, 5.05048056648896935105798953309, 6.72938513523557232140948636734, 8.480180020560308628827573910100, 10.05812064764471585634015974276, 10.48401421374017222106396617180, 12.38426576723532720731866696087, 13.58105318479496894879637000843, 14.78804696078371171633596728878, 15.61802693998778086390230173733

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