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Properties

Label 2-21e2-7.4-c3-0-19
Degree $2$
Conductor $441$
Sign $0.605 + 0.795i$
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  + (−1.5 − 2.59i)2-s + (−0.5 + 0.866i)4-s + (1.5 + 2.59i)5-s − 21·8-s + (4.5 − 7.79i)10-s + (−7.5 + 12.9i)11-s + 64·13-s + (35.5 + 61.4i)16-s + (−42 + 72.7i)17-s + (−8 − 13.8i)19-s − 3.00·20-s + 45·22-s + (−42 − 72.7i)23-s + (58 − 100. i)25-s + (−96 − 166. i)26-s + ⋯
L(s)  = 1  + (−0.530 − 0.918i)2-s + (−0.0625 + 0.108i)4-s + (0.134 + 0.232i)5-s − 0.928·8-s + (0.142 − 0.246i)10-s + (−0.205 + 0.356i)11-s + 1.36·13-s + (0.554 + 0.960i)16-s + (−0.599 + 1.03i)17-s + (−0.0965 − 0.167i)19-s − 0.0335·20-s + 0.436·22-s + (−0.380 − 0.659i)23-s + (0.464 − 0.803i)25-s + (−0.724 − 1.25i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(26.0198\)
Root analytic conductor: \(5.10096\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{441} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.451656811\)
\(L(\frac12)\) \(\approx\) \(1.451656811\)
\(L(\frac{5}{2})\) 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 \( 1 + (1.5 + 2.59i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-1.5 - 2.59i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (7.5 - 12.9i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 64T + 2.19e3T^{2} \)
17 \( 1 + (42 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (8 + 13.8i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (42 + 72.7i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 297T + 2.43e4T^{2} \)
31 \( 1 + (126.5 - 219. i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-158 - 273. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 360T + 6.89e4T^{2} \)
43 \( 1 - 26T + 7.95e4T^{2} \)
47 \( 1 + (-15 - 25.9i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-181.5 + 314. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (-7.5 + 12.9i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (59 + 102. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-185 + 320. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 342T + 3.57e5T^{2} \)
73 \( 1 + (-181 + 313. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (233.5 + 404. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 477T + 5.71e5T^{2} \)
89 \( 1 + (453 + 784. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + 503T + 9.12e5T^{2} \)
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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

−10.60982644287911418758961908882, −9.973218517059621998057799642773, −8.777504961730791300813302332479, −8.302807898883038704078072954677, −6.62527404408692767441820137638, −6.08138782445709783735751988771, −4.52550649009489103195520527079, −3.23648330000092306167548948425, −2.12193959654287018804306110627, −0.920477360918718739067526671656, 0.77496663956091385349843618612, 2.67373217283019854892413570213, 3.98766008989132867855067248652, 5.49575018011669787358588336015, 6.22841474261099443111631796615, 7.22791433762193334569516139643, 8.077111420380656843208190387772, 8.910746689049461258871995135307, 9.519499412395209552185180453650, 10.88592742798339387609587095604

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