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Properties

Label 1-37-37.33-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: $\chi_{37} (33, \cdot )$
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

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

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