Properties

Label 1-37-37.23-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: $\chi_{37} (23, \cdot )$
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

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

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