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Properties

Label 1-6e2-36.11-r0-0-0
Degree $1$
Conductor $36$
Sign $0.984 - 0.173i$
Analytic cond. $0.167183$
Root an. cond. $0.167183$
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  + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s − 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + 35-s + 37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)5-s + (0.5 + 0.866i)7-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s − 17-s − 19-s + (−0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (0.5 + 0.866i)29-s + (0.5 − 0.866i)31-s + 35-s + 37-s + (0.5 − 0.866i)41-s + (0.5 + 0.866i)43-s + (−0.5 − 0.866i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(36\)    =    \(2^{2} \cdot 3^{2}\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(0.167183\)
Root analytic conductor: \(0.167183\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{36} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 36,\ (0:\ ),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8222820649 - 0.07194035890i\)
\(L(\frac12)\) \(\approx\) \(0.8222820649 - 0.07194035890i\)
\(L(1)\) \(\approx\) \(1.003837166 - 0.05783774636i\)
\(L(1)\) \(\approx\) \(1.003837166 - 0.05783774636i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 - T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 - T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−36.00525677501136349756372020648, −34.32559475708389148454345625691, −33.59985372037602719440421343101, −32.410377116432291659288084035111, −30.709460580858837860147969780235, −30.00072573157095372216919796400, −28.70783783763675238999960959202, −27.137992170147889144606728655683, −26.19713634504464078781560156483, −24.94746180270863493885699970595, −23.41316604449680327709302166251, −22.36886154269478372052879267445, −20.96480270317158436691406526239, −19.73252620095592696928190534912, −18.050841047001815437271326540424, −17.29453928386547325791063122901, −15.34094671625210298882315145436, −14.22256159954633033580274083309, −12.85216184693912363313832213407, −10.91146184578109010597210076996, −10.01002603795048827922562440232, −7.881531411731270516392470116776, −6.5261700824006092798773189112, −4.55075499877039430801681565928, −2.41511530935627412534947040506, 2.14409489942574708228920090773, 4.676075508748655458490821851313, 6.06203944896421696376204364589, 8.278342933690077090023168122797, 9.35390753944101506765847727587, 11.25279823282029673209891658310, 12.648325869272542811061859051250, 13.97502385043088870723119736977, 15.58071229483743724297034263903, 16.889791040762615134076772847028, 18.169039935900764697177121392176, 19.586675629076211234362596893133, 21.18031806028247547499431076424, 21.79998852975647480543378572569, 23.870700345723942497068926763896, 24.59162420280926346667433058825, 25.93193174639440572944941739710, 27.43679296486427561565187471753, 28.57261072631808318262395139322, 29.5081392852163109266857295296, 31.27588267842812972985599510952, 31.980390955335435978492680125656, 33.43701017854223800184568857670, 34.48935673580116156526442706543, 35.86690603875864830574326672474

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