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Properties

Label 1-37-37.5-r1-0-0
Degree $1$
Conductor $37$
Sign $0.213 + 0.977i$
Analytic cond. $3.97620$
Root an. cond. $3.97620$
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  + (−0.642 − 0.766i)2-s + (−0.766 − 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + i·6-s + (−0.939 + 0.342i)7-s + (0.866 − 0.5i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.766 − 0.642i)12-s + (0.984 + 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)2-s + (−0.766 − 0.642i)3-s + (−0.173 + 0.984i)4-s + (−0.342 − 0.939i)5-s + i·6-s + (−0.939 + 0.342i)7-s + (0.866 − 0.5i)8-s + (0.173 + 0.984i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.766 − 0.642i)12-s + (0.984 + 0.173i)13-s + (0.866 + 0.5i)14-s + (−0.342 + 0.939i)15-s + (−0.939 − 0.342i)16-s + (−0.984 + 0.173i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.213 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.213 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.213 + 0.977i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ 0.213 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1083672477 + 0.08726646715i\)
\(L(\frac12)\) \(\approx\) \(0.1083672477 + 0.08726646715i\)
\(L(1)\) \(\approx\) \(0.3832365656 - 0.1718731866i\)
\(L(1)\) \(\approx\) \(0.3832365656 - 0.1718731866i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
good2 \( 1 + (-0.642 - 0.766i)T \)
3 \( 1 + (-0.766 - 0.642i)T \)
5 \( 1 + (-0.342 - 0.939i)T \)
7 \( 1 + (-0.939 + 0.342i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.984 + 0.173i)T \)
17 \( 1 + (-0.984 + 0.173i)T \)
19 \( 1 + (-0.642 + 0.766i)T \)
23 \( 1 + (-0.866 - 0.5i)T \)
29 \( 1 + (-0.866 + 0.5i)T \)
31 \( 1 - iT \)
41 \( 1 + (-0.173 + 0.984i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (0.342 - 0.939i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (0.939 - 0.342i)T \)
71 \( 1 + (0.766 + 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.342 - 0.939i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (-0.342 + 0.939i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−34.9878245385283754204293970665, −33.99536434134035299953751387736, −32.9589002499881259487406769407, −32.03850904932065915603768218632, −29.94464507181257486098812776037, −28.76810500550162904909982751713, −27.565861172149412919583947658543, −26.57053287049967456267051514182, −25.78860146912035121721735002177, −23.90886664755232648673333157845, −22.90061152868788105013763474816, −21.98565570510731059408672399177, −19.838781538214457984883660195934, −18.61801572486747399031173766379, −17.38819153159372671708803747011, −16.09411274743672813331450749093, −15.370071905514547620028052535, −13.71468426871066802157244758139, −11.27580538890110694348443363863, −10.38736269672151304711249796815, −8.93800585932748263264698459317, −6.86504443277927639152419135198, −6.005816163948271956414787896913, −3.828297684997818824321245700780, −0.12352703136094995655490453035, 1.71803990947925726334750393171, 4.21308672734253584048682741904, 6.41436033326693304967003882318, 8.15585646400261687354723655729, 9.56497575420961305840147532240, 11.24863188243077488114564566937, 12.45996389156278132856097994937, 13.09939674151563620498986171858, 16.02779647559662477994237500788, 16.95435636353644813164491998510, 18.24592577238899128191072039293, 19.38831634440378432213658248056, 20.45467546055101580175638251983, 22.10424812561150464917726995043, 23.153176227630249852833368903952, 24.74131724185713264304390485980, 25.89138897326668345693189037677, 27.79984496189184211004951901853, 28.31053915616562671010813054117, 29.27821071182184299726585815754, 30.53067085196272276371660012087, 31.657876353526719133551699401572, 33.33352707285324685364141098237, 35.084831428031147269693551720102, 35.59157530839088270117937176927

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