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Properties

Label 2-11e2-11.5-c1-0-1
Degree $2$
Conductor $121$
Sign $0.944 - 0.329i$
Analytic cond. $0.966189$
Root an. cond. $0.982949$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)3-s + (1.61 − 1.17i)4-s + (−0.927 + 2.85i)5-s + (−0.618 − 1.90i)9-s + 2·12-s + (−2.42 + 1.76i)15-s + (1.23 − 3.80i)16-s + (1.85 + 5.70i)20-s − 9·23-s + (−3.23 − 2.35i)25-s + (1.54 − 4.75i)27-s + (−1.54 − 4.75i)31-s + (−3.23 − 2.35i)36-s + (−5.66 + 4.11i)37-s + 6.00·45-s + ⋯
L(s)  = 1  + (0.467 + 0.339i)3-s + (0.809 − 0.587i)4-s + (−0.414 + 1.27i)5-s + (−0.206 − 0.634i)9-s + 0.577·12-s + (−0.626 + 0.455i)15-s + (0.309 − 0.951i)16-s + (0.414 + 1.27i)20-s − 1.87·23-s + (−0.647 − 0.470i)25-s + (0.297 − 0.915i)27-s + (−0.277 − 0.854i)31-s + (−0.539 − 0.391i)36-s + (−0.931 + 0.676i)37-s + 0.894·45-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.944 - 0.329i$
Analytic conductor: \(0.966189\)
Root analytic conductor: \(0.982949\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 121,\ (\ :1/2),\ 0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.26359 + 0.214398i\)
\(L(\frac12)\) \(\approx\) \(1.26359 + 0.214398i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (0.927 - 2.85i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 9T + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.54 + 4.75i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (5.66 - 4.11i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-9.70 - 7.05i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-12.1 + 8.81i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 13T + 67T^{2} \)
71 \( 1 + (0.927 - 2.85i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-5.25 - 16.1i)T + (-78.4 + 57.0i)T^{2} \)
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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

−14.03703770718823490918072935071, −12.18765629536355270337707450895, −11.34388967657658285121754372024, −10.41982794356555770475955770534, −9.571647589286248238059675020654, −7.986189174377125712027807869998, −6.85246772846024821651366745207, −5.92954515821149489802277856239, −3.80119557959838207945530827534, −2.54351762753205675592695845243, 2.06610799285378510264236344863, 3.85811350682600829681631682069, 5.43391847772256698767773231700, 7.10172663489530703404904093521, 8.147067558637862565197103207270, 8.698767798797060376802557990818, 10.35937683398210052435572657562, 11.66495924215872650497682043028, 12.38016300511332446403407496821, 13.21869544586834940052979605289

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