Properties

Label 1-11e2-121.76-r1-0-0
Degree $1$
Conductor $121$
Sign $0.985 - 0.167i$
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  + (0.142 − 0.989i)2-s + 3-s + (−0.959 − 0.281i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (−0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + 9-s + (0.959 − 0.281i)10-s + (−0.959 − 0.281i)12-s + (0.959 + 0.281i)13-s + (0.415 + 0.909i)14-s + (0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (0.654 − 0.755i)17-s + (0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s + 3-s + (−0.959 − 0.281i)4-s + (0.415 + 0.909i)5-s + (0.142 − 0.989i)6-s + (−0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + 9-s + (0.959 − 0.281i)10-s + (−0.959 − 0.281i)12-s + (0.959 + 0.281i)13-s + (0.415 + 0.909i)14-s + (0.415 + 0.909i)15-s + (0.841 + 0.540i)16-s + (0.654 − 0.755i)17-s + (0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.985 - 0.167i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(121\)    =    \(11^{2}\)
Sign: $0.985 - 0.167i$
Analytic conductor: \(13.0032\)
Root analytic conductor: \(13.0032\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{121} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 121,\ (1:\ ),\ 0.985 - 0.167i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.458328807 - 0.2079315125i\)
\(L(\frac12)\) \(\approx\) \(2.458328807 - 0.2079315125i\)
\(L(1)\) \(\approx\) \(1.517962834 - 0.3157273338i\)
\(L(1)\) \(\approx\) \(1.517962834 - 0.3157273338i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
good2 \( 1 + (0.142 - 0.989i)T \)
3 \( 1 + T \)
5 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (0.959 + 0.281i)T \)
17 \( 1 + (0.654 - 0.755i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.841 + 0.540i)T \)
29 \( 1 + (0.654 + 0.755i)T \)
31 \( 1 + (-0.959 + 0.281i)T \)
37 \( 1 + (-0.959 + 0.281i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (0.841 - 0.540i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (0.142 + 0.989i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.841 - 0.540i)T \)
79 \( 1 + (-0.415 - 0.909i)T \)
83 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−28.68025587592171808422901020251, −27.56860487211531502834476017152, −26.37991741519674812398356566022, −25.7181315904305724612437345029, −24.95687356400038277751460952959, −24.00741730639857267808504665793, −23.02517029650132618380419077999, −21.6656959778264876471812574595, −20.697312297462894014039269003368, −19.61721456132726931425684756190, −18.48453526715293703537109514985, −17.18358396704704431376798576514, −16.21984025596327462769852424363, −15.41433185343483190565369027587, −14.07374269161603376093429787275, −13.279198089231338131919665542165, −12.64769477702237375657946019885, −10.16571361556752541571849602564, −9.1722948768146305641854687451, −8.3540149709343616698884559955, −7.14608673818206391464611087545, −5.871433011762108125944938951880, −4.40339279333977088157300492050, −3.270491349626930010918672005009, −0.989501636866102466167661561564, 1.60898165732750300073867256965, 2.99900830123238683222933749111, 3.53730502168108244027384709511, 5.5208864019144918074086336438, 7.05363909400118583329903778123, 8.70832554265717875985341445378, 9.60463595212720406821348846772, 10.47208651955912467101349128942, 11.88032102456472091662032443118, 13.13441151424403239952494332276, 13.93173393008761041025572883706, 14.82453634278122714128469932867, 16.10762371722244271002375136942, 18.10897703455282097583968496731, 18.70476443710084867462382081167, 19.472568308392008108313940770457, 20.69381712033675757255339071761, 21.46258738786391533889309367163, 22.4394298280221957790578869780, 23.36958007473561138712125843949, 25.09696515283148699997818473904, 25.8254587923566427082119691464, 26.76597033660325223303894502627, 27.752200777911497584776326851046, 29.18703364088582000861724220924

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