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Properties

Label 1-37-37.29-r1-0-0
Degree $1$
Conductor $37$
Sign $-0.881 + 0.471i$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s − i·6-s + (−0.5 − 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)3-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s − i·6-s + (−0.5 − 0.866i)7-s − i·8-s + (−0.5 + 0.866i)9-s + 10-s − 11-s + (−0.5 + 0.866i)12-s + (−0.866 + 0.5i)13-s + i·14-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $-0.881 + 0.471i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ -0.881 + 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09053571291 + 0.3609777886i\)
\(L(\frac12)\) \(\approx\) \(0.09053571291 + 0.3609777886i\)
\(L(1)\) \(\approx\) \(0.5117635123 + 0.1549053842i\)
\(L(1)\) \(\approx\) \(0.5117635123 + 0.1549053842i\)

Euler product

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

−34.9066528272182748422424713341, −34.2664252735897850896988000063, −32.06244533698453825966401095974, −31.60795876488632470076245681358, −29.78682647071996046851146011617, −28.646196199413833750062426602047, −27.54638552764291938485488694739, −26.133635499869826101349533419327, −25.13040913425457295541449611357, −24.13644713481389304125848592437, −23.126285210517339395797933289601, −20.71377993510570818896521313163, −19.41528793437835292556803549603, −18.82854987735978636697766598597, −17.42523841987185044704441592555, −15.825254312647143323758725999466, −14.91450952323888946058735694712, −12.94725202815197280802513293547, −11.68970330135087068399409197331, −9.57824227482554863985948981713, −8.259581094013353617601059207090, −7.37105026768783361230159224211, −5.58252641741673014664747258603, −2.579042834589445045057661803713, −0.28571526045412949858901243672, 2.85855265856409634715555517549, 4.142512282580021924346953429121, 7.22402228956581453922790526349, 8.401096949666111721503734398547, 10.124678819231345667460929064096, 10.74968617119897751410260966260, 12.52926978284387350920693022684, 14.52280935518386128980969109728, 15.96938847173357718457398406582, 16.855246086112261092638360446, 18.81644238302818460813813070232, 19.68252818987094256134887775834, 20.7445538228875487608040231283, 22.00430910090601822485661219181, 23.4718722596492762351504810891, 25.5645122581727911814006034141, 26.52232107219536141045531880564, 27.08287362332175127631816440008, 28.37755120256851534303190719761, 29.75197791992480951706470429763, 30.98910836944265197884761088301, 32.03900869609211142328223506665, 33.73649918244985268010184616003, 34.60845765912475310626563227203, 36.20746601875277781328829846227

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