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Properties

Label 2-21e2-21.5-c3-0-33
Degree $2$
Conductor $441$
Sign $0.681 + 0.731i$
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  + (3.10 + 1.79i)2-s + (2.44 + 4.22i)4-s + (0.428 − 0.742i)5-s − 11.1i·8-s + (2.66 − 1.53i)10-s + (−0.321 + 0.185i)11-s − 62.0i·13-s + (39.6 − 68.6i)16-s + (−51.9 − 90.0i)17-s + (−61.9 − 35.7i)19-s + 4.18·20-s − 1.33·22-s + (118. + 68.2i)23-s + (62.1 + 107. i)25-s + (111. − 192. i)26-s + ⋯
L(s)  = 1  + (1.09 + 0.634i)2-s + (0.305 + 0.528i)4-s + (0.0383 − 0.0664i)5-s − 0.494i·8-s + (0.0842 − 0.0486i)10-s + (−0.00880 + 0.00508i)11-s − 1.32i·13-s + (0.618 − 1.07i)16-s + (−0.741 − 1.28i)17-s + (−0.747 − 0.431i)19-s + 0.0468·20-s − 0.0129·22-s + (1.07 + 0.618i)23-s + (0.497 + 0.860i)25-s + (0.839 − 1.45i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.681 + 0.731i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (215, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.681 + 0.731i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.885378490\)
\(L(\frac12)\) \(\approx\) \(2.885378490\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.10 - 1.79i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + (-0.428 + 0.742i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (0.321 - 0.185i)T + (665.5 - 1.15e3i)T^{2} \)
13 \( 1 + 62.0iT - 2.19e3T^{2} \)
17 \( 1 + (51.9 + 90.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (61.9 + 35.7i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-118. - 68.2i)T + (6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 17.8iT - 2.43e4T^{2} \)
31 \( 1 + (93.2 - 53.8i)T + (1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-190. + 330. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 - 101.T + 6.89e4T^{2} \)
43 \( 1 + 326.T + 7.95e4T^{2} \)
47 \( 1 + (-261. + 452. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-268. + 154. i)T + (7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (174. + 302. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-580. - 335. i)T + (1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-119. - 207. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 178. iT - 3.57e5T^{2} \)
73 \( 1 + (-117. + 67.7i)T + (1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (428. - 742. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 1.24e3T + 5.71e5T^{2} \)
89 \( 1 + (282. - 489. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.73e3iT - 9.12e5T^{2} \)
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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

−10.73132346382708783464792421289, −9.633064123346603796532477902165, −8.727629127440667625681677119117, −7.38857067340635527542535577609, −6.82444992492378611932328595142, −5.53484999626172643212533652695, −5.05558779431418440148534816428, −3.83717399392071256975225874569, −2.71737160468718185913614196690, −0.64402482240549505337938100855, 1.69963444058151126767412217670, 2.77271587371986348297865215811, 4.11276383288760137118282190347, 4.60935203306412708808041544097, 5.99377378556261977594150499724, 6.75788646317302151931168484098, 8.252731432975078865842582229893, 8.950769790800172327323683196210, 10.29614520817555411698300070101, 11.05621938246506122997850211251

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