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Properties

Label 2-21e2-7.2-c3-0-9
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  + (−0.207 + 0.358i)2-s + (3.91 + 6.77i)4-s + (0.0502 − 0.0870i)5-s − 6.55·8-s + (0.0208 + 0.0360i)10-s + (−21.9 − 38.0i)11-s + 16.6·13-s + (−29.9 + 51.8i)16-s + (60.8 + 105. i)17-s + (−63.5 + 110. i)19-s + 0.786·20-s + 18.2·22-s + (26.7 − 46.4i)23-s + (62.4 + 108. i)25-s + (−3.44 + 5.97i)26-s + ⋯
L(s)  = 1  + (−0.0732 + 0.126i)2-s + (0.489 + 0.847i)4-s + (0.00449 − 0.00778i)5-s − 0.289·8-s + (0.000658 + 0.00114i)10-s + (−0.602 − 1.04i)11-s + 0.355·13-s + (−0.468 + 0.810i)16-s + (0.867 + 1.50i)17-s + (−0.767 + 1.32i)19-s + 0.00879·20-s + 0.176·22-s + (0.242 − 0.420i)23-s + (0.499 + 0.865i)25-s + (−0.0260 + 0.0450i)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

\(L(2)\) \(\approx\) \(1.251766484\)
\(L(\frac12)\) \(\approx\) \(1.251766484\)
\(L(\frac{5}{2})\) 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 \( 1 + (0.207 - 0.358i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (-0.0502 + 0.0870i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (21.9 + 38.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 - 16.6T + 2.19e3T^{2} \)
17 \( 1 + (-60.8 - 105. i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (63.5 - 110. i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-26.7 + 46.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 235.T + 2.43e4T^{2} \)
31 \( 1 + (9.35 + 16.2i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-95.9 + 166. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 319.T + 6.89e4T^{2} \)
43 \( 1 + 218.T + 7.95e4T^{2} \)
47 \( 1 + (200. - 347. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (-321. - 556. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-5.80 - 10.0i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (6.12 - 10.6i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (334. + 579. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 822.T + 3.57e5T^{2} \)
73 \( 1 + (-257. - 446. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (-402. + 697. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + 394.T + 5.71e5T^{2} \)
89 \( 1 + (336. - 583. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 - 1.09e3T + 9.12e5T^{2} \)
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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

−10.95029443923321722818483700694, −10.45708574225980530635815264326, −9.003621902060222507516681642473, −8.199528884845955452211005041157, −7.62896441071531748233403601459, −6.32181010519490488642911067889, −5.63772738394136881441678953044, −3.91719241024404697659632781407, −3.18629329963424325766841295994, −1.68328282566716467142245798787, 0.38889937368521281850529144697, 1.89353054178358461777811314164, 2.98664271962340096130436673436, 4.74015876659192877055192795973, 5.42619903762497557994698507791, 6.74750940058676248173332926419, 7.30633880186448245478213409996, 8.655251188662727090448071908926, 9.712787899594252403946370191931, 10.20913898668624654313867404006

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