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Properties

Label 2-21e2-7.4-c3-0-23
Degree $2$
Conductor $441$
Sign $-0.266 - 0.963i$
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 + 3.46i)2-s + (−3.99 + 6.92i)4-s + (2 + 3.46i)5-s + (−7.99 + 13.8i)10-s + (31 − 53.6i)11-s + 62·13-s + (31.9 + 55.4i)16-s + (−42 + 72.7i)17-s + (50 + 86.6i)19-s − 31.9·20-s + 248·22-s + (−21 − 36.3i)23-s + (54.5 − 94.3i)25-s + (124 + 214. i)26-s + 10·29-s + ⋯
L(s)  = 1  + (0.707 + 1.22i)2-s + (−0.499 + 0.866i)4-s + (0.178 + 0.309i)5-s + (−0.252 + 0.438i)10-s + (0.849 − 1.47i)11-s + 1.32·13-s + (0.499 + 0.866i)16-s + (−0.599 + 1.03i)17-s + (0.603 + 1.04i)19-s − 0.357·20-s + 2.40·22-s + (−0.190 − 0.329i)23-s + (0.435 − 0.755i)25-s + (0.935 + 1.62i)26-s + 0.0640·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.266 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(441\)    =    \(3^{2} \cdot 7^{2}\)
Sign: $-0.266 - 0.963i$
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: \((2,\ 441,\ (\ :3/2),\ -0.266 - 0.963i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.400168828\)
\(L(\frac12)\) \(\approx\) \(3.400168828\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2 - 3.46i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + (-2 - 3.46i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-31 + 53.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 62T + 2.19e3T^{2} \)
17 \( 1 + (42 - 72.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-50 - 86.6i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (21 + 36.3i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 - 10T + 2.43e4T^{2} \)
31 \( 1 + (24 - 41.5i)T + (-1.48e4 - 2.57e4i)T^{2} \)
37 \( 1 + (-123 - 213. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 248T + 6.89e4T^{2} \)
43 \( 1 - 68T + 7.95e4T^{2} \)
47 \( 1 + (162 + 280. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-129 + 223. i)T + (-7.44e4 - 1.28e5i)T^{2} \)
59 \( 1 + (60 - 103. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-311 - 538. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (452 - 782. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 678T + 3.57e5T^{2} \)
73 \( 1 + (321 - 555. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (370 + 640. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 468T + 5.71e5T^{2} \)
89 \( 1 + (100 + 173. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 1.26e3T + 9.12e5T^{2} \)
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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

−10.97243618838578852741091962799, −10.17374828020073240013522114154, −8.483697254092750268115934490116, −8.408040075433781663016324565477, −6.86993161531219033184036837264, −6.18687543893505468745118794139, −5.67499072268838903026670057578, −4.18982102974614174638649916255, −3.40961292006127990460344089987, −1.34367975921383036261734921342, 1.07237533859292919610992333894, 2.12097258348363533111229685037, 3.39893247367747519061621449493, 4.41862198501957200140632077060, 5.17827915420782659295821722332, 6.62719138379382806629512412425, 7.58581335285119630267708267129, 9.167653644842767042435496421138, 9.533804085030187635325435411250, 10.79140244527796179132987238589

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