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Properties

Label 1-11e2-121.25-r0-0-0
Degree $1$
Conductor $121$
Sign $0.998 - 0.0519i$
Analytic cond. $0.561921$
Root an. cond. $0.561921$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.564 + 0.825i)2-s + (−0.809 + 0.587i)3-s + (−0.362 − 0.931i)4-s + (−0.921 − 0.389i)5-s + (−0.0285 − 0.999i)6-s + (−0.870 + 0.491i)7-s + (0.974 + 0.226i)8-s + (0.309 − 0.951i)9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (0.774 − 0.633i)13-s + (0.0855 − 0.996i)14-s + (0.974 − 0.226i)15-s + (−0.736 + 0.676i)16-s + (0.897 − 0.441i)17-s + (0.610 + 0.791i)18-s + ⋯
L(s)  = 1  + (−0.564 + 0.825i)2-s + (−0.809 + 0.587i)3-s + (−0.362 − 0.931i)4-s + (−0.921 − 0.389i)5-s + (−0.0285 − 0.999i)6-s + (−0.870 + 0.491i)7-s + (0.974 + 0.226i)8-s + (0.309 − 0.951i)9-s + (0.841 − 0.540i)10-s + (0.841 + 0.540i)12-s + (0.774 − 0.633i)13-s + (0.0855 − 0.996i)14-s + (0.974 − 0.226i)15-s + (−0.736 + 0.676i)16-s + (0.897 − 0.441i)17-s + (0.610 + 0.791i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0519i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.998 - 0.0519i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.998 - 0.0519i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3844587951 + 0.009984168580i\)
\(L(\frac12)\) \(\approx\) \(0.3844587951 + 0.009984168580i\)
\(L(1)\) \(\approx\) \(0.4670954296 + 0.1316608842i\)
\(L(1)\) \(\approx\) \(0.4670954296 + 0.1316608842i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (-0.564 + 0.825i)T \)
3 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 + (-0.921 - 0.389i)T \)
7 \( 1 + (-0.870 + 0.491i)T \)
13 \( 1 + (0.774 - 0.633i)T \)
17 \( 1 + (0.897 - 0.441i)T \)
19 \( 1 + (-0.466 - 0.884i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (0.696 - 0.717i)T \)
31 \( 1 + (-0.254 - 0.967i)T \)
37 \( 1 + (-0.998 - 0.0570i)T \)
41 \( 1 + (0.941 - 0.336i)T \)
43 \( 1 + (-0.654 - 0.755i)T \)
47 \( 1 + (0.610 - 0.791i)T \)
53 \( 1 + (-0.736 - 0.676i)T \)
59 \( 1 + (0.941 + 0.336i)T \)
61 \( 1 + (-0.564 - 0.825i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.985 + 0.170i)T \)
73 \( 1 + (0.198 - 0.980i)T \)
79 \( 1 + (0.516 + 0.856i)T \)
83 \( 1 + (0.993 + 0.113i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
97 \( 1 + (-0.921 + 0.389i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.04010613041875416815088452621, −28.17844496212075642612500332414, −27.28714128816844913158377140509, −26.28112841546144609464855237586, −25.25512371784092467170513467905, −23.49578800532772439056865295189, −23.06066679587700510317600980735, −22.0539800405457116412229870829, −20.76610874760663031087850416680, −19.36536421924571929131340588354, −19.0245299780314891760878457874, −18.00692876292226461642833573844, −16.62233366690479316469311616717, −16.17879382997270259120254100122, −14.11957674746274798451035586342, −12.73996544689225909702991115825, −12.14822718003555128527840133691, −10.920968841674624265143188378935, −10.29349698896810138658765363718, −8.559530728113507993804275931744, −7.4030072800382034087614182562, −6.40498039228349992500777387919, −4.33550571537569061933391078010, −3.15799934973019870910267171126, −1.233767449489422916390852384444, 0.57821782563149301902579621083, 3.6154628954756435559725914337, 5.04479282468580884699353816511, 6.02620934351280393026857587779, 7.26164852761393499934139467986, 8.65722441789715498198503519566, 9.6268058259239288062518631860, 10.81334997249204191600503477956, 11.97099067960474302021643137052, 13.26880587974071827655510348402, 15.16200500321783779835451821623, 15.67133620006349517533520618382, 16.445826342894038911829983522786, 17.44846471780019101080306381379, 18.64776896617089414607085487281, 19.542093663994259128706126519937, 20.84926639905584775953851495419, 22.37091236177450790081080175701, 23.12439271593625761296813317637, 23.81406927286320390254488376888, 25.134759700612702716237950798248, 26.09205120530790876707910564453, 27.1998412499986457092609642518, 27.95385244409829390190442565812, 28.47266897197024886237749021153

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