Properties

Label 2-11e2-11.5-c3-0-21
Degree $2$
Conductor $121$
Sign $-0.944 + 0.329i$
Analytic cond. $7.13923$
Root an. cond. $2.67193$
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  + (−6.47 − 4.70i)3-s + (6.47 − 4.70i)4-s + (5.56 − 17.1i)5-s + (11.4 + 35.1i)9-s − 64·12-s + (−116. + 84.6i)15-s + (19.7 − 60.8i)16-s + (−44.4 − 136. i)20-s − 108·23-s + (−160. − 116. i)25-s + (24.7 − 76.0i)27-s + (105. + 323. i)31-s + (239. + 173. i)36-s + (351. − 255. i)37-s + 666·45-s + ⋯
L(s)  = 1  + (−1.24 − 0.904i)3-s + (0.809 − 0.587i)4-s + (0.497 − 1.53i)5-s + (0.423 + 1.30i)9-s − 1.53·12-s + (−2.00 + 1.45i)15-s + (0.309 − 0.951i)16-s + (−0.497 − 1.53i)20-s − 0.979·23-s + (−1.28 − 0.935i)25-s + (0.176 − 0.542i)27-s + (0.608 + 1.87i)31-s + (1.10 + 0.805i)36-s + (1.56 − 1.13i)37-s + 2.20·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(7.13923\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :3/2),\ -0.944 + 0.329i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.201844 - 1.18960i\)
\(L(\frac12)\) \(\approx\) \(0.201844 - 1.18960i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-6.47 + 4.70i)T^{2} \)
3 \( 1 + (6.47 + 4.70i)T + (8.34 + 25.6i)T^{2} \)
5 \( 1 + (-5.56 + 17.1i)T + (-101. - 73.4i)T^{2} \)
7 \( 1 + (105. - 326. i)T^{2} \)
13 \( 1 + (-1.77e3 + 1.29e3i)T^{2} \)
17 \( 1 + (-3.97e3 - 2.88e3i)T^{2} \)
19 \( 1 + (2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + 108T + 1.21e4T^{2} \)
29 \( 1 + (7.53e3 - 2.31e4i)T^{2} \)
31 \( 1 + (-105. - 323. i)T + (-2.41e4 + 1.75e4i)T^{2} \)
37 \( 1 + (-351. + 255. i)T + (1.56e4 - 4.81e4i)T^{2} \)
41 \( 1 + (2.12e4 + 6.55e4i)T^{2} \)
43 \( 1 + 7.95e4T^{2} \)
47 \( 1 + (-29.1 - 21.1i)T + (3.20e4 + 9.87e4i)T^{2} \)
53 \( 1 + (228. + 701. i)T + (-1.20e5 + 8.75e4i)T^{2} \)
59 \( 1 + (-582. + 423. i)T + (6.34e4 - 1.95e5i)T^{2} \)
61 \( 1 + (-1.83e5 - 1.33e5i)T^{2} \)
67 \( 1 + 416T + 3.00e5T^{2} \)
71 \( 1 + (-189. + 582. i)T + (-2.89e5 - 2.10e5i)T^{2} \)
73 \( 1 + (1.20e5 - 3.69e5i)T^{2} \)
79 \( 1 + (-3.98e5 + 2.89e5i)T^{2} \)
83 \( 1 + (-4.62e5 - 3.36e5i)T^{2} \)
89 \( 1 - 1.67e3T + 7.04e5T^{2} \)
97 \( 1 + (10.5 + 32.3i)T + (-7.38e5 + 5.36e5i)T^{2} \)
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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

−12.39013762170305326752225823079, −11.73192697499522264578332450728, −10.65235589622745417810869971896, −9.472763262618668507949373388747, −7.985445085622730871184504535243, −6.63569402112610197700212531436, −5.77514625800853500204102325643, −4.92532901675372698611808546685, −1.80673807592092051780541294447, −0.73736457287548767861729506349, 2.59746914012761409600697914585, 4.06522403580169364017072692867, 5.90503619536109020927685983675, 6.49318848123488723063215350588, 7.74610184048241485852764267269, 9.803193952001627024248566536668, 10.48125300160710025761438972615, 11.33587029423411038573644578691, 11.86357081068329006659452105624, 13.39287271661242716528364267142

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