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Properties

Label 2-21e2-7.2-c3-0-45
Degree $2$
Conductor $441$
Sign $-0.827 - 0.561i$
Analytic cond. $26.0198$
Root an. cond. $5.10096$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.26 − 3.92i)2-s + (−6.26 − 10.8i)4-s + (6.73 − 11.6i)5-s − 20.5·8-s + (−30.5 − 52.8i)10-s + (0.406 + 0.704i)11-s − 34.9·13-s + (3.60 − 6.25i)16-s + (−58.8 − 101. i)17-s + (−46.6 + 80.7i)19-s − 168.·20-s + 3.68·22-s + (60.1 − 104. i)23-s + (−28.3 − 49.0i)25-s + (−79.1 + 137. i)26-s + ⋯
L(s)  = 1  + (0.800 − 1.38i)2-s + (−0.783 − 1.35i)4-s + (0.602 − 1.04i)5-s − 0.907·8-s + (−0.965 − 1.67i)10-s + (0.0111 + 0.0193i)11-s − 0.745·13-s + (0.0564 − 0.0976i)16-s + (−0.839 − 1.45i)17-s + (−0.563 + 0.975i)19-s − 1.88·20-s + 0.0357·22-s + (0.545 − 0.944i)23-s + (−0.226 − 0.392i)25-s + (−0.597 + 1.03i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.827 - 0.561i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ -0.827 - 0.561i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.551372228\)
\(L(\frac12)\) \(\approx\) \(2.551372228\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-2.26 + 3.92i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-6.73 + 11.6i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (-0.406 - 0.704i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 34.9T + 2.19e3T^{2} \)
17 \( 1 + (58.8 + 101. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (46.6 - 80.7i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-60.1 + 104. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 8.56T + 2.43e4T^{2} \)
31 \( 1 + (41.0 + 71.1i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (14.4 - 24.9i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 70.5T + 6.89e4T^{2} \)
43 \( 1 - 417.T + 7.95e4T^{2} \)
47 \( 1 + (169. - 292. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-74.5 - 129. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-47.0 - 81.5i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (60.2 - 104. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-396. - 686. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 449.T + 3.57e5T^{2} \)
73 \( 1 + (234. + 406. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-509. + 883. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 104.T + 5.71e5T^{2} \)
89 \( 1 + (-786. + 1.36e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 550.T + 9.12e5T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.30766423852883519897582447220, −9.501840245614067885605835833804, −8.797719595280668033338245338460, −7.37096744936873520279972325545, −5.90666807656564708634939004665, −4.89671743852820899502076095008, −4.35377466797817829458240878885, −2.83999427450375027508499936254, −1.88290798214248062184794840050, −0.60270185352272281651558128632, 2.20594695440074016721310206189, 3.59801557699201928618041007033, 4.75087752196605273539928204625, 5.75859828061992322377839051547, 6.62223567401324972656868673451, 7.10610052405819103099480747946, 8.196698446146649349141697329961, 9.233276766247545498902403847047, 10.43376934220117160303223275185, 11.10880273665892622821773513336

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