Properties

Label 2-11e2-11.9-c3-0-16
Degree $2$
Conductor $121$
Sign $0.263 + 0.964i$
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  + (0.844 − 2.59i)2-s + (6.41 − 4.66i)3-s + (0.433 + 0.314i)4-s + (4.59 + 14.1i)5-s + (−6.69 − 20.6i)6-s + (−2.48 − 1.80i)7-s + (18.8 − 13.7i)8-s + (11.0 − 34.1i)9-s + 40.5·10-s + 4.24·12-s + (1.65 − 5.09i)13-s + (−6.78 + 4.93i)14-s + (95.2 + 69.2i)15-s + (−18.3 − 56.5i)16-s + (−12.7 − 39.1i)17-s + (−79.2 − 57.5i)18-s + ⋯
L(s)  = 1  + (0.298 − 0.918i)2-s + (1.23 − 0.896i)3-s + (0.0541 + 0.0393i)4-s + (0.410 + 1.26i)5-s + (−0.455 − 1.40i)6-s + (−0.134 − 0.0974i)7-s + (0.833 − 0.605i)8-s + (0.410 − 1.26i)9-s + 1.28·10-s + 0.102·12-s + (0.0353 − 0.108i)13-s + (−0.129 + 0.0941i)14-s + (1.64 + 1.19i)15-s + (−0.286 − 0.883i)16-s + (−0.181 − 0.559i)17-s + (−1.03 − 0.753i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.263 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.263 + 0.964i$
Analytic conductor: \(7.13923\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :3/2),\ 0.263 + 0.964i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.46149 - 1.87938i\)
\(L(\frac12)\) \(\approx\) \(2.46149 - 1.87938i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-0.844 + 2.59i)T + (-6.47 - 4.70i)T^{2} \)
3 \( 1 + (-6.41 + 4.66i)T + (8.34 - 25.6i)T^{2} \)
5 \( 1 + (-4.59 - 14.1i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (2.48 + 1.80i)T + (105. + 326. i)T^{2} \)
13 \( 1 + (-1.65 + 5.09i)T + (-1.77e3 - 1.29e3i)T^{2} \)
17 \( 1 + (12.7 + 39.1i)T + (-3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (113. - 82.2i)T + (2.11e3 - 6.52e3i)T^{2} \)
23 \( 1 + 111.T + 1.21e4T^{2} \)
29 \( 1 + (-20.2 - 14.6i)T + (7.53e3 + 2.31e4i)T^{2} \)
31 \( 1 + (-9.73 + 29.9i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (10.6 + 7.72i)T + (1.56e4 + 4.81e4i)T^{2} \)
41 \( 1 + (211. - 153. i)T + (2.12e4 - 6.55e4i)T^{2} \)
43 \( 1 + 57.7T + 7.95e4T^{2} \)
47 \( 1 + (-278. + 202. i)T + (3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (105. - 326. i)T + (-1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (71.4 + 51.9i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-228. - 702. i)T + (-1.83e5 + 1.33e5i)T^{2} \)
67 \( 1 - 342.T + 3.00e5T^{2} \)
71 \( 1 + (64.0 + 197. i)T + (-2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + (-817. - 594. i)T + (1.20e5 + 3.69e5i)T^{2} \)
79 \( 1 + (-399. + 1.23e3i)T + (-3.98e5 - 2.89e5i)T^{2} \)
83 \( 1 + (-136. - 420. i)T + (-4.62e5 + 3.36e5i)T^{2} \)
89 \( 1 + 1.48e3T + 7.04e5T^{2} \)
97 \( 1 + (-416. + 1.28e3i)T + (-7.38e5 - 5.36e5i)T^{2} \)
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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

−12.87340780168848320051476586131, −11.90456148528071177529396365863, −10.68114751231208988396781805975, −9.896941002780388417716387381361, −8.343858179995238018671254971401, −7.28418639893016841405687652406, −6.42904795448833773161805050568, −3.79922818328960131024151482187, −2.72493225577288563438372508662, −1.90292313021274514585857725482, 2.11045685231451057465266713155, 4.14578433025132730272360736925, 5.07983402338143609667550359041, 6.44144481101255645971048402620, 8.105107901654657528821668284395, 8.730754738675673910906612230944, 9.708350173648780799391513524727, 10.84925496254287239616619798227, 12.60360799669340347379088979168, 13.58403469515996753470443045009

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