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Properties

Label 1-11e2-121.103-r0-0-0
Degree $1$
Conductor $121$
Sign $0.704 + 0.709i$
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  + (0.774 + 0.633i)2-s + (−0.809 − 0.587i)3-s + (0.198 + 0.980i)4-s + (0.897 + 0.441i)5-s + (−0.254 − 0.967i)6-s + (0.0855 − 0.996i)7-s + (−0.466 + 0.884i)8-s + (0.309 + 0.951i)9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (0.993 + 0.113i)13-s + (0.696 − 0.717i)14-s + (−0.466 − 0.884i)15-s + (−0.921 + 0.389i)16-s + (−0.564 + 0.825i)17-s + (−0.362 + 0.931i)18-s + ⋯
L(s)  = 1  + (0.774 + 0.633i)2-s + (−0.809 − 0.587i)3-s + (0.198 + 0.980i)4-s + (0.897 + 0.441i)5-s + (−0.254 − 0.967i)6-s + (0.0855 − 0.996i)7-s + (−0.466 + 0.884i)8-s + (0.309 + 0.951i)9-s + (0.415 + 0.909i)10-s + (0.415 − 0.909i)12-s + (0.993 + 0.113i)13-s + (0.696 − 0.717i)14-s + (−0.466 − 0.884i)15-s + (−0.921 + 0.389i)16-s + (−0.564 + 0.825i)17-s + (−0.362 + 0.931i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.704 + 0.709i$
Analytic conductor: \(0.561921\)
Root analytic conductor: \(0.561921\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (0:\ ),\ 0.704 + 0.709i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.315246453 + 0.5477959042i\)
\(L(\frac12)\) \(\approx\) \(1.315246453 + 0.5477959042i\)
\(L(1)\) \(\approx\) \(1.322514500 + 0.3811844752i\)
\(L(1)\) \(\approx\) \(1.322514500 + 0.3811844752i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.774 + 0.633i)T \)
3 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (0.897 + 0.441i)T \)
7 \( 1 + (0.0855 - 0.996i)T \)
13 \( 1 + (0.993 + 0.113i)T \)
17 \( 1 + (-0.564 + 0.825i)T \)
19 \( 1 + (0.941 + 0.336i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (0.610 - 0.791i)T \)
31 \( 1 + (-0.736 + 0.676i)T \)
37 \( 1 + (-0.870 - 0.491i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (-0.142 - 0.989i)T \)
47 \( 1 + (-0.362 - 0.931i)T \)
53 \( 1 + (-0.921 - 0.389i)T \)
59 \( 1 + (-0.998 + 0.0570i)T \)
61 \( 1 + (0.774 - 0.633i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (0.974 + 0.226i)T \)
79 \( 1 + (-0.985 + 0.170i)T \)
83 \( 1 + (0.516 + 0.856i)T \)
89 \( 1 + (-0.959 - 0.281i)T \)
97 \( 1 + (0.897 - 0.441i)T \)
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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

−28.895216992246496017693909095385, −28.22288247879766717354905702486, −27.44621415292873258099843940304, −25.74105953892364689934848914721, −24.62039345102379341208162081790, −23.72058878942526710436900688454, −22.421572590585700817802095475727, −21.90357377326259479604700169593, −20.99997781525541837551277532107, −20.21943383238047898458620519900, −18.42572547822875411281876027669, −17.82492908183054408937213009012, −16.157273826694939143287595518343, −15.50067567183418502161237592565, −14.08955045891217131320060626992, −13.0173742955475690431212830133, −11.93031234850330524787432074090, −11.094571702667158243967762511705, −9.807630234155885484734926272285, −9.035727085818809282015090597293, −6.440975112346704170573427777089, −5.552607947491287380599646367974, −4.78791382717571965401016870530, −3.150179676834616291586357092787, −1.501252191987432170452110401905, 1.83085396695526153646578121560, 3.72610793505749124039679817291, 5.19689563422525395582917640247, 6.29962290602228492216796749350, 6.96782706507911966235467289357, 8.2917687894527642691808985949, 10.30954623875358098764113391698, 11.25832492316479219381580933810, 12.61089057944487460711035214429, 13.61587913770643336313576918315, 14.13741293884266665395092940491, 15.79870191928181104365018998912, 16.845678951992877579441224915604, 17.59613531288428586991641929531, 18.46989247547321646319447203158, 20.22506677377429823240295533447, 21.401706962360707235928277967728, 22.34734094483269321588103667363, 23.137357812354725955626677364502, 24.00248515048730469352856763928, 24.92755608738725815152431759439, 25.96766370064271635875512444822, 26.84395278367936676850436496172, 28.53181668953129789900688314564, 29.37311183022554523984620398540

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