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Properties

Label 1-37-37.12-r0-0-0
Degree $1$
Conductor $37$
Sign $0.721 - 0.691i$
Analytic cond. $0.171827$
Root an. cond. $0.171827$
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  + (0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + 6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)2-s + (0.173 + 0.984i)3-s + (−0.939 − 0.342i)4-s + (0.766 − 0.642i)5-s + 6-s + (0.766 − 0.642i)7-s + (−0.5 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.173 − 0.984i)12-s + (−0.939 − 0.342i)13-s + (−0.5 − 0.866i)14-s + (0.766 + 0.642i)15-s + (0.766 + 0.642i)16-s + (−0.939 + 0.342i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.721 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.721 - 0.691i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (12, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ 0.721 - 0.691i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8250224266 - 0.3315152153i\)
\(L(\frac12)\) \(\approx\) \(0.8250224266 - 0.3315152153i\)
\(L(1)\) \(\approx\) \(1.025206045 - 0.3186274895i\)
\(L(1)\) \(\approx\) \(1.025206045 - 0.3186274895i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
good2 \( 1 + (0.173 - 0.984i)T \)
3 \( 1 + (0.173 + 0.984i)T \)
5 \( 1 + (0.766 - 0.642i)T \)
7 \( 1 + (0.766 - 0.642i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.939 + 0.342i)T \)
19 \( 1 + (0.173 + 0.984i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + T \)
41 \( 1 + (-0.939 - 0.342i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.766 + 0.642i)T \)
59 \( 1 + (0.766 + 0.642i)T \)
61 \( 1 + (-0.939 - 0.342i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 + 0.984i)T \)
73 \( 1 + T \)
79 \( 1 + (0.766 - 0.642i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (0.766 + 0.642i)T \)
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

−35.55737244664605154613137169247, −34.2809305672477336427861243052, −33.89980765779648276451667289294, −32.09234708713787103886376632073, −31.093774336178057767399289335626, −30.018781440877329512780321028275, −28.684025902822820084645160046464, −26.8296953501195039182700183313, −25.848418219523711037517215212360, −24.59056865419119869131220675196, −24.087985943873179471615206080660, −22.42432591963739819022554320830, −21.38763882005088796796970839023, −19.13023842138606415053999947674, −18.0458315857800586178253031838, −17.33007217366417523049812313020, −15.33804110826129334285532790950, −14.11945426123335035977092320659, −13.30528742196747246570942608615, −11.557289986060654190407696866, −9.23745036491563480214768809695, −7.86234835456625411321924754634, −6.55728926928468506751583600239, −5.31307204237638369371586457279, −2.54193907413823288034334573757, 2.17733868424224259379051329304, 4.3054459372668812777970979105, 5.2340206278592525882109227640, 8.37999505948536789603704018279, 9.833202932226007530465870431, 10.598219017762581822571447456715, 12.33723625771391243172553405852, 13.80586156663537205214950836896, 14.91440701781358440790082633920, 16.92263581474839209067959109801, 17.90049769163092449931166644606, 20.06845691166944875742896842296, 20.59269378898605207965879493732, 21.64437545092678502072254924358, 22.795143236752151450274487423455, 24.39524085695291235329320005029, 26.1691517186216148658972919316, 27.27337132409410490147818940341, 28.297910379631695360519916849738, 29.30528880362276735912024146544, 30.76022308794907941365770727132, 31.857772195123667980447237104573, 32.93643948050253003917803524281, 33.75713042285888893353109045744, 36.071765343100618653714908989790

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