Properties

Label 1-11e2-121.80-r0-0-0
Degree $1$
Conductor $121$
Sign $-0.152 - 0.988i$
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  + (−0.254 − 0.967i)2-s + (−0.809 + 0.587i)3-s + (−0.870 + 0.491i)4-s + (−0.985 + 0.170i)5-s + (0.774 + 0.633i)6-s + (0.974 − 0.226i)7-s + (0.696 + 0.717i)8-s + (0.309 − 0.951i)9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (−0.736 − 0.676i)13-s + (−0.466 − 0.884i)14-s + (0.696 − 0.717i)15-s + (0.516 − 0.856i)16-s + (−0.0285 − 0.999i)17-s + (−0.998 − 0.0570i)18-s + ⋯
L(s)  = 1  + (−0.254 − 0.967i)2-s + (−0.809 + 0.587i)3-s + (−0.870 + 0.491i)4-s + (−0.985 + 0.170i)5-s + (0.774 + 0.633i)6-s + (0.974 − 0.226i)7-s + (0.696 + 0.717i)8-s + (0.309 − 0.951i)9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (−0.736 − 0.676i)13-s + (−0.466 − 0.884i)14-s + (0.696 − 0.717i)15-s + (0.516 − 0.856i)16-s + (−0.0285 − 0.999i)17-s + (−0.998 − 0.0570i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.152 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.152 - 0.988i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (80, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ -0.152 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3416768138 - 0.3984782970i\)
\(L(\frac12)\) \(\approx\) \(0.3416768138 - 0.3984782970i\)
\(L(1)\) \(\approx\) \(0.5523547275 - 0.2580757826i\)
\(L(1)\) \(\approx\) \(0.5523547275 - 0.2580757826i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.254 + 0.967i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.985 - 0.170i)T \)
7 \( 1 + (-0.974 + 0.226i)T \)
13 \( 1 + (0.736 + 0.676i)T \)
17 \( 1 + (0.0285 + 0.999i)T \)
19 \( 1 + (-0.610 + 0.791i)T \)
23 \( 1 + (0.654 + 0.755i)T \)
29 \( 1 + (-0.941 - 0.336i)T \)
31 \( 1 + (-0.993 + 0.113i)T \)
37 \( 1 + (-0.198 + 0.980i)T \)
41 \( 1 + (0.362 - 0.931i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (0.998 - 0.0570i)T \)
53 \( 1 + (-0.516 - 0.856i)T \)
59 \( 1 + (0.362 + 0.931i)T \)
61 \( 1 + (0.254 - 0.967i)T \)
67 \( 1 + (-0.841 + 0.540i)T \)
71 \( 1 + (0.564 + 0.825i)T \)
73 \( 1 + (-0.0855 - 0.996i)T \)
79 \( 1 + (-0.897 - 0.441i)T \)
83 \( 1 + (0.921 + 0.389i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (0.985 + 0.170i)T \)
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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

−29.05158456506500395613594529413, −28.01016798732310681306838259495, −27.40563318411715886724878234819, −26.44350596708551387778938294051, −24.91192074114493419178130369220, −24.16262808293689034057968849053, −23.623794752719707021396596434015, −22.63466201171309111524741873422, −21.50309119167984123841670440881, −19.6446103695895940948517188539, −18.84622718726758925282736524924, −17.80328941072809657110751825383, −16.96753945473805378988875773699, −15.98712880101742757502363378790, −14.91550308296171805333845937378, −13.802264101074041283838968701331, −12.32182937987099560158692298974, −11.51718663563461857340779679647, −10.10481841425301385111045948064, −8.31870435373036669449841220644, −7.7307145496623132858997155671, −6.54326047841344280668169786051, −5.23747093627714548910467011092, −4.286892641077362355141876437387, −1.43325435237717705098739884206, 0.68286394556516333799056509170, 2.91130424986653206875725330513, 4.36796967418061160023805276316, 5.04610483835662400756713109162, 7.24387861199449902375887845237, 8.42086868349999307040844253178, 9.86456813726525415047200175283, 10.86677193511196771499859644028, 11.655294838672648320201414030946, 12.37191032304943392299987804628, 14.08602082804170248162150157787, 15.30894308909994073535602737305, 16.50464900184599871789431068298, 17.682873393772740237156330339284, 18.32245149180193507724467438774, 19.82719264379723364146720038698, 20.512012485210336956612650626302, 21.654446960883474860728183057, 22.56317359610066672607973576099, 23.32942484695988658257796732434, 24.47305114885090236246375376981, 26.54160638416058939984242264308, 27.00018175399518713425289255200, 27.7718439213849439004872909177, 28.5679190232794381723577278989

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