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Properties

Label 2-11e2-11.6-c2-0-0
Degree $2$
Conductor $121$
Sign $-0.917 + 0.396i$
Analytic cond. $3.29701$
Root an. cond. $1.81576$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.34 + 0.437i)2-s + (0.809 + 0.587i)3-s + (−1.61 + 1.17i)4-s + (−2.16 + 6.65i)5-s + (−1.34 − 0.437i)6-s + (−4.15 − 5.72i)7-s + (4.98 − 6.86i)8-s + (−2.47 − 7.60i)9-s − 9.89i·10-s − 1.99·12-s + (−16.1 + 5.24i)13-s + (8.09 + 5.87i)14-s + (−5.66 + 4.11i)15-s + (−1.23 + 3.80i)16-s + (−4.03 − 1.31i)17-s + (6.65 + 9.15i)18-s + ⋯
L(s)  = 1  + (−0.672 + 0.218i)2-s + (0.269 + 0.195i)3-s + (−0.404 + 0.293i)4-s + (−0.432 + 1.33i)5-s + (−0.224 − 0.0728i)6-s + (−0.593 − 0.817i)7-s + (0.623 − 0.858i)8-s + (−0.274 − 0.845i)9-s − 0.989i·10-s − 0.166·12-s + (−1.24 + 0.403i)13-s + (0.577 + 0.419i)14-s + (−0.377 + 0.274i)15-s + (−0.0772 + 0.237i)16-s + (−0.237 − 0.0771i)17-s + (0.369 + 0.508i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.917 + 0.396i$
Analytic conductor: \(3.29701\)
Root analytic conductor: \(1.81576\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1),\ -0.917 + 0.396i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0351090 - 0.169640i\)
\(L(\frac12)\) \(\approx\) \(0.0351090 - 0.169640i\)
\(L(2)\) 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 \( 1 + (1.34 - 0.437i)T + (3.23 - 2.35i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T + (2.78 + 8.55i)T^{2} \)
5 \( 1 + (2.16 - 6.65i)T + (-20.2 - 14.6i)T^{2} \)
7 \( 1 + (4.15 + 5.72i)T + (-15.1 + 46.6i)T^{2} \)
13 \( 1 + (16.1 - 5.24i)T + (136. - 99.3i)T^{2} \)
17 \( 1 + (4.03 + 1.31i)T + (233. + 169. i)T^{2} \)
19 \( 1 + (9.97 - 13.7i)T + (-111. - 343. i)T^{2} \)
23 \( 1 + 9T + 529T^{2} \)
29 \( 1 + (13.3 + 18.3i)T + (-259. + 799. i)T^{2} \)
31 \( 1 + (-15.1 - 46.6i)T + (-777. + 564. i)T^{2} \)
37 \( 1 + (13.7 - 9.99i)T + (423. - 1.30e3i)T^{2} \)
41 \( 1 + (9.97 - 13.7i)T + (-519. - 1.59e3i)T^{2} \)
43 \( 1 - 46.6iT - 1.84e3T^{2} \)
47 \( 1 + (25.8 + 18.8i)T + (682. + 2.10e3i)T^{2} \)
53 \( 1 + (-4.94 - 15.2i)T + (-2.27e3 + 1.65e3i)T^{2} \)
59 \( 1 + (-57.4 + 41.7i)T + (1.07e3 - 3.31e3i)T^{2} \)
61 \( 1 + (-10.7 - 3.49i)T + (3.01e3 + 2.18e3i)T^{2} \)
67 \( 1 + 31T + 4.48e3T^{2} \)
71 \( 1 + (22.5 - 69.4i)T + (-4.07e3 - 2.96e3i)T^{2} \)
73 \( 1 + (-23.2 - 32.0i)T + (-1.64e3 + 5.06e3i)T^{2} \)
79 \( 1 + (149. - 48.5i)T + (5.04e3 - 3.66e3i)T^{2} \)
83 \( 1 + (33.6 + 10.9i)T + (5.57e3 + 4.04e3i)T^{2} \)
89 \( 1 + 9T + 7.92e3T^{2} \)
97 \( 1 + (5.25 + 16.1i)T + (-7.61e3 + 5.53e3i)T^{2} \)
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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

−14.06903931724162637590396382966, −12.77593989000245066493230477683, −11.64212366697186704674615295765, −10.22325209087656157297016358708, −9.787559739862359271488581725952, −8.438305042958634816348792528991, −7.25431312949564746636462178193, −6.61457048744240397936463492626, −4.18224555385823010787947088520, −3.16715813033563837891221390844, 0.14149476369867150733504856545, 2.24345540222685757515894333542, 4.63300019448663646711160891405, 5.53610822990112509631253874804, 7.62588332176574310980242653422, 8.604060321149626375017188312824, 9.193507334010383194173488928012, 10.29337316366306772901053461410, 11.65194751986545513546832969426, 12.71651598537071100564635597292

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