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Properties

Label 1-11e2-121.74-r1-0-0
Degree $1$
Conductor $121$
Sign $-0.709 - 0.704i$
Analytic cond. $13.0032$
Root an. cond. $13.0032$
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

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

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $-0.709 - 0.704i$
Analytic conductor: \(13.0032\)
Root analytic conductor: \(13.0032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (74, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (1:\ ),\ -0.709 - 0.704i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.05186054802 - 0.1258952746i\)
\(L(\frac12)\) \(\approx\) \(0.05186054802 - 0.1258952746i\)
\(L(1)\) \(\approx\) \(0.5041045395 + 0.09048638961i\)
\(L(1)\) \(\approx\) \(0.5041045395 + 0.09048638961i\)

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

−29.33426225372283519349606869566, −28.23912040999804196467218554838, −27.54824352662666077590058705379, −26.156267364809933561663660744851, −25.177437860210729745686899168058, −24.48320513908630193330545051653, −22.71182560978801137191636597929, −21.952778571125559109606646333113, −21.232203473211713807101533469973, −19.65531458433767145055719702085, −18.61475976732495463533865629532, −18.06143981060784069302754364032, −17.14098914363280342145376532815, −16.13893447051351898532653651514, −14.43463029693343748286852720339, −12.89696716770682184458715911744, −12.24535716568443676845769157361, −11.07625979615190294395053629938, −10.07940556909228260559862734145, −8.982907911424632441202345819087, −7.47254081209047021635200180138, −6.3889990964149051489163466284, −5.09123576441183927363699113597, −2.737929468452975338938548655290, −1.802828711085409745615052534512, 0.07716738995720312210060727426, 1.623380584526344920024232821258, 4.26783084603811969627719714108, 5.51601766396277516554323478065, 6.41991157938420089866419554708, 7.75367274089924700060635243870, 9.376651156067770123333561995541, 10.05610209647377552210844380039, 10.918713381591259177476124234556, 12.52772089334246058501369554062, 13.997679763212911479338747061638, 15.07096621883979915938093871274, 16.39202595929061961712283376782, 17.15549635098287091986121040942, 17.47039838074463382675853743919, 18.97793256873204751935775198524, 20.24624021585620154562694063002, 21.23879881304922634068178719716, 22.443285494449332541431976718830, 23.628685341634925472392261441369, 24.2748534867151912509429086787, 25.68390086403239166168898743518, 26.38177440166326864452541228562, 27.49755490705100662071045234720, 28.165193553568939524424097149015

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