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Properties

Label 2-11e2-11.8-c4-0-14
Degree $2$
Conductor $121$
Sign $0.822 - 0.568i$
Analytic cond. $12.5077$
Root an. cond. $3.53663$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.16 + 6.65i)3-s + (4.94 − 15.2i)4-s + (39.6 + 28.8i)5-s + (25.8 − 18.8i)9-s + 112·12-s + (−105. + 326. i)15-s + (−207. − 150. i)16-s + (634. − 460. i)20-s + 167·23-s + (548. + 1.68e3i)25-s + (639. + 464. i)27-s + (447. − 325. i)31-s + (−158. − 486. i)36-s + (−652. + 2.00e3i)37-s + 1.56e3·45-s + ⋯
L(s)  = 1  + (0.240 + 0.739i)3-s + (0.309 − 0.951i)4-s + (1.58 + 1.15i)5-s + (0.319 − 0.232i)9-s + 0.777·12-s + (−0.471 + 1.44i)15-s + (−0.809 − 0.587i)16-s + (1.58 − 1.15i)20-s + 0.315·23-s + (0.878 + 2.70i)25-s + (0.877 + 0.637i)27-s + (0.465 − 0.338i)31-s + (−0.122 − 0.375i)36-s + (−0.476 + 1.46i)37-s + 0.774·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.822 - 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.822 - 0.568i$
Analytic conductor: \(12.5077\)
Root analytic conductor: \(3.53663\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :2),\ 0.822 - 0.568i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.59982 + 0.810696i\)
\(L(\frac12)\) \(\approx\) \(2.59982 + 0.810696i\)
\(L(3)\) 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 \( 1 + (-4.94 + 15.2i)T^{2} \)
3 \( 1 + (-2.16 - 6.65i)T + (-65.5 + 47.6i)T^{2} \)
5 \( 1 + (-39.6 - 28.8i)T + (193. + 594. i)T^{2} \)
7 \( 1 + (1.94e3 + 1.41e3i)T^{2} \)
13 \( 1 + (-8.82e3 + 2.71e4i)T^{2} \)
17 \( 1 + (-2.58e4 - 7.94e4i)T^{2} \)
19 \( 1 + (1.05e5 - 7.66e4i)T^{2} \)
23 \( 1 - 167T + 2.79e5T^{2} \)
29 \( 1 + (5.72e5 + 4.15e5i)T^{2} \)
31 \( 1 + (-447. + 325. i)T + (2.85e5 - 8.78e5i)T^{2} \)
37 \( 1 + (652. - 2.00e3i)T + (-1.51e6 - 1.10e6i)T^{2} \)
41 \( 1 + (2.28e6 - 1.66e6i)T^{2} \)
43 \( 1 - 3.41e6T^{2} \)
47 \( 1 + (592. + 1.82e3i)T + (-3.94e6 + 2.86e6i)T^{2} \)
53 \( 1 + (-580. + 422. i)T + (2.43e6 - 7.50e6i)T^{2} \)
59 \( 1 + (-1.38e3 + 4.26e3i)T + (-9.80e6 - 7.12e6i)T^{2} \)
61 \( 1 + (-4.27e6 - 1.31e7i)T^{2} \)
67 \( 1 + 7.75e3T + 2.01e7T^{2} \)
71 \( 1 + (6.15e3 + 4.47e3i)T + (7.85e6 + 2.41e7i)T^{2} \)
73 \( 1 + (2.29e7 + 1.66e7i)T^{2} \)
79 \( 1 + (-1.20e7 + 3.70e7i)T^{2} \)
83 \( 1 + (-1.46e7 - 4.51e7i)T^{2} \)
89 \( 1 + 6.43e3T + 6.27e7T^{2} \)
97 \( 1 + (-7.92e3 + 5.75e3i)T + (2.73e7 - 8.41e7i)T^{2} \)
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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

−13.22476096929726438994497079363, −11.42006520530438214705683390517, −10.29930429273660776952921668259, −10.03547444870615627874498436445, −9.097782403809155465503942227097, −6.94263537458860499624928686805, −6.17447509414084752984516589471, −5.00224359355394836781927393807, −3.04803536578489198867736507910, −1.67606184532287877163390867852, 1.39907161383751665500969806756, 2.51483272952586915999501499643, 4.60218986933411740910503413302, 6.01234409029463020188725595383, 7.21242436895936195126782496138, 8.386513777028267083013285172282, 9.231874710121139807696179309065, 10.47577003273415870877211447475, 12.08074406715214533026502025251, 12.85566746106703414531683425037

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