Properties

Label 2-11e2-11.4-c3-0-22
Degree $2$
Conductor $121$
Sign $-0.927 - 0.374i$
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  + (2.47 − 7.60i)3-s + (−2.47 − 7.60i)4-s + (−14.5 + 10.5i)5-s + (−29.9 − 21.7i)9-s − 63.9·12-s + (44.4 + 136. i)15-s + (−51.7 + 37.6i)16-s + (116. + 84.6i)20-s − 108·23-s + (61.4 − 189. i)25-s + (−64.7 + 47.0i)27-s + (−275. − 199. i)31-s + (−91.4 + 281. i)36-s + (−134. − 412. i)37-s + 665.·45-s + ⋯
L(s)  = 1  + (0.475 − 1.46i)3-s + (−0.309 − 0.951i)4-s + (−1.30 + 0.946i)5-s + (−1.10 − 0.805i)9-s − 1.53·12-s + (0.765 + 2.35i)15-s + (−0.809 + 0.587i)16-s + (1.30 + 0.946i)20-s − 0.979·23-s + (0.491 − 1.51i)25-s + (−0.461 + 0.335i)27-s + (−1.59 − 1.15i)31-s + (−0.423 + 1.30i)36-s + (−0.595 − 1.83i)37-s + 2.20·45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.927 - 0.374i$
Analytic conductor: \(7.13923\)
Root analytic conductor: \(2.67193\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :3/2),\ -0.927 - 0.374i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.122359 + 0.629957i\)
\(L(\frac12)\) \(\approx\) \(0.122359 + 0.629957i\)
\(L(\frac{5}{2})\) 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 \( 1 + (2.47 + 7.60i)T^{2} \)
3 \( 1 + (-2.47 + 7.60i)T + (-21.8 - 15.8i)T^{2} \)
5 \( 1 + (14.5 - 10.5i)T + (38.6 - 118. i)T^{2} \)
7 \( 1 + (-277. + 201. i)T^{2} \)
13 \( 1 + (678. + 2.08e3i)T^{2} \)
17 \( 1 + (1.51e3 - 4.67e3i)T^{2} \)
19 \( 1 + (-5.54e3 - 4.03e3i)T^{2} \)
23 \( 1 + 108T + 1.21e4T^{2} \)
29 \( 1 + (-1.97e4 + 1.43e4i)T^{2} \)
31 \( 1 + (275. + 199. i)T + (9.20e3 + 2.83e4i)T^{2} \)
37 \( 1 + (134. + 412. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
41 \( 1 + (-5.57e4 - 4.05e4i)T^{2} \)
43 \( 1 + 7.95e4T^{2} \)
47 \( 1 + (11.1 - 34.2i)T + (-8.39e4 - 6.10e4i)T^{2} \)
53 \( 1 + (-597. - 433. i)T + (4.60e4 + 1.41e5i)T^{2} \)
59 \( 1 + (222. + 684. i)T + (-1.66e5 + 1.20e5i)T^{2} \)
61 \( 1 + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + 416T + 3.00e5T^{2} \)
71 \( 1 + (495. - 359. i)T + (1.10e5 - 3.40e5i)T^{2} \)
73 \( 1 + (-3.14e5 + 2.28e5i)T^{2} \)
79 \( 1 + (1.52e5 + 4.68e5i)T^{2} \)
83 \( 1 + (1.76e5 - 5.43e5i)T^{2} \)
89 \( 1 - 1.67e3T + 7.04e5T^{2} \)
97 \( 1 + (-27.5 - 19.9i)T + (2.82e5 + 8.68e5i)T^{2} \)
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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

−12.43410282574236512228058169936, −11.46653023355564881838776983441, −10.50941905211301725348524041876, −8.962539283011321833536927259661, −7.74778074866152736214190724862, −7.07981167763666819667030106055, −5.91050881802025028260150712366, −3.87143943570272122254363997370, −2.15067441588553204296212581698, −0.31473449184664528444840498007, 3.35508336762068117440483655075, 4.13539684171945595860860964164, 5.00154385944659121018435388445, 7.48632012191810004029955355413, 8.533591763917543177240355199418, 9.002458197999781371364427809406, 10.33187672895483970520223321846, 11.63726980580672560569529253569, 12.35236261169854758437274672620, 13.54063908174915817164876329754

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