Properties

Label 1-37-37.9-r0-0-0
Degree $1$
Conductor $37$
Sign $0.308 + 0.951i$
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  + (−0.939 + 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)5-s + 6-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.939 + 0.342i)12-s + (0.766 − 0.642i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯
L(s)  = 1  + (−0.939 + 0.342i)2-s + (−0.939 − 0.342i)3-s + (0.766 − 0.642i)4-s + (0.173 + 0.984i)5-s + 6-s + (0.173 + 0.984i)7-s + (−0.5 + 0.866i)8-s + (0.766 + 0.642i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.939 + 0.342i)12-s + (0.766 − 0.642i)13-s + (−0.5 − 0.866i)14-s + (0.173 − 0.984i)15-s + (0.173 − 0.984i)16-s + (0.766 + 0.642i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.308 + 0.951i$
Analytic conductor: \(0.171827\)
Root analytic conductor: \(0.171827\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (0:\ ),\ 0.308 + 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3387689976 + 0.2463383707i\)
\(L(\frac12)\) \(\approx\) \(0.3387689976 + 0.2463383707i\)
\(L(1)\) \(\approx\) \(0.5141135679 + 0.1890252611i\)
\(L(1)\) \(\approx\) \(0.5141135679 + 0.1890252611i\)

Euler product

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

−35.6887437987906309473449624788, −34.19940628961324891287922706823, −33.37311616375265696587040745674, −31.963849351652988132695554601335, −29.94856402991601843407874976128, −29.17669371075645523393012330836, −28.10120443336542996952337605100, −27.2363455694531834197012671416, −26.03400678622807338094633865136, −24.35503559927700417729462166358, −23.31253900040740223194064362602, −21.27746505023538404053698380273, −20.803180065205761685583900115294, −19.10922210113540284288694781904, −17.663289239140107375950063270300, −16.662256572756819511352614686177, −16.00880985175202842300528627355, −13.3611792889233864929033344085, −11.85204881748222339407736477112, −10.73387647780722427993386688733, −9.4834900758039433466025178343, −7.889672769997696131266974599839, −6.04335502022936829237954956842, −4.08693912453885005046630802124, −1.066304661875605489943296197327, 2.16828332359103987012630255802, 5.58565877500320855220537128698, 6.65682600505255515202025980711, 8.11615554756115148830681122744, 10.11578384186285172454652874287, 11.045658761844004527984171109582, 12.51272272056808278721842915668, 14.8019834745334814965051567656, 15.84988659186725624705732229180, 17.49134801425802534891942423009, 18.24601397087965543223215313148, 19.09195681799801524974163430816, 21.10955477505210088732818220551, 22.600915360103759940431336488499, 23.72081432421016735845086913540, 25.16528623931479587676886814123, 26.057842203623865231350734315714, 27.76651440284003970988088648420, 28.27627158125074995379056915446, 29.66897908557935071753221542111, 30.63457114835901680782399670290, 32.81047192013317073968927420897, 34.085200455595879000421782554704, 34.48464826512778759308484506444, 35.5908434801810988213770576543

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