Properties

Label 2-21e2-7.2-c3-0-47
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  + (2.65 − 4.59i)2-s + (−10.0 − 17.4i)4-s + (−2.78 + 4.81i)5-s − 64.6·8-s + (14.7 + 25.5i)10-s + (−6.95 − 12.0i)11-s − 38.6·13-s + (−90.8 + 157. i)16-s + (−21.7 − 37.6i)17-s + (−54.5 + 94.4i)19-s + 112.·20-s − 73.8·22-s + (−37.4 + 64.8i)23-s + (47.0 + 81.4i)25-s + (−102. + 177. i)26-s + ⋯
L(s)  = 1  + (0.938 − 1.62i)2-s + (−1.26 − 2.18i)4-s + (−0.248 + 0.430i)5-s − 2.85·8-s + (0.466 + 0.808i)10-s + (−0.190 − 0.330i)11-s − 0.825·13-s + (−1.41 + 2.45i)16-s + (−0.310 − 0.537i)17-s + (−0.658 + 1.14i)19-s + 1.25·20-s − 0.715·22-s + (−0.339 + 0.587i)23-s + (0.376 + 0.651i)25-s + (−0.774 + 1.34i)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: $\chi_{441} (226, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 441,\ (\ :3/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.07682458403\)
\(L(\frac12)\) \(\approx\) \(0.07682458403\)
\(L(\frac{5}{2})\) 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 \( 1 + (-2.65 + 4.59i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + (2.78 - 4.81i)T + (-62.5 - 108. i)T^{2} \)
11 \( 1 + (6.95 + 12.0i)T + (-665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 38.6T + 2.19e3T^{2} \)
17 \( 1 + (21.7 + 37.6i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (54.5 - 94.4i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (37.4 - 64.8i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 - 72.3T + 2.43e4T^{2} \)
31 \( 1 + (-32.0 - 55.4i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (94.3 - 163. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + 24.7T + 6.89e4T^{2} \)
43 \( 1 + 243.T + 7.95e4T^{2} \)
47 \( 1 + (-310. + 537. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + (143. + 249. i)T + (-7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (262. + 454. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (191. - 332. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (99.0 + 171. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 785.T + 3.57e5T^{2} \)
73 \( 1 + (165. + 286. i)T + (-1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (218. - 379. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 241.T + 5.71e5T^{2} \)
89 \( 1 + (792. - 1.37e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 79.2T + 9.12e5T^{2} \)
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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

−10.27242316282823050596639763208, −9.533694655190003406619735227916, −8.351993193804418590144551512443, −6.90956343324848268491132223786, −5.66302659499996476961097933087, −4.76927284604221326219772686261, −3.68318422021775237385144339894, −2.81756153933936493034272681450, −1.65221613018659013127719043014, −0.01729676698181807466941693272, 2.73216010362568910415006088164, 4.33403753550246576633681050488, 4.69115469685437902619954024268, 5.89025262390120226405936049478, 6.76307387627932289189216753165, 7.58434200727932744721983163177, 8.450445184870413445095199356000, 9.177476826975389689986942028176, 10.55846842212557921620287929137, 11.98415854596321033401882014133

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