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Properties

Label 1-37-37.20-r1-0-0
Degree $1$
Conductor $37$
Sign $0.849 - 0.527i$
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  + (−0.342 − 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (0.984 − 0.173i)5-s − i·6-s + (0.173 + 0.984i)7-s + (0.866 + 0.5i)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 + (−0.642 − 0.766i)13-s + (0.866 − 0.5i)14-s + (0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (0.939 + 0.342i)3-s + (−0.766 + 0.642i)4-s + (0.984 − 0.173i)5-s − i·6-s + (0.173 + 0.984i)7-s + (0.866 + 0.5i)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 + (−0.642 − 0.766i)13-s + (0.866 − 0.5i)14-s + (0.984 + 0.173i)15-s + (0.173 − 0.984i)16-s + (0.642 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.849 - 0.527i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(37\)
Sign: $0.849 - 0.527i$
Analytic conductor: \(3.97620\)
Root analytic conductor: \(3.97620\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{37} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 37,\ (1:\ ),\ 0.849 - 0.527i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.714553287 - 0.4889529956i\)
\(L(\frac12)\) \(\approx\) \(1.714553287 - 0.4889529956i\)
\(L(1)\) \(\approx\) \(1.300246394 - 0.3269880849i\)
\(L(1)\) \(\approx\) \(1.300246394 - 0.3269880849i\)

Euler product

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

−35.70990568920022313900163533121, −34.07959662089613002437938300285, −33.00851632768363959313807596241, −32.245420880207470150719237859736, −30.68861621783860156842864679504, −29.58787380744392077567555787156, −27.91978332901180730660063223355, −26.29124555671612959509602557596, −25.942761790289076974691017621418, −24.63420108569863615530313606870, −23.68960146411522367265449549110, −22.00340840838448803907929524395, −20.367464425202374521032042040350, −19.1247861552966700373168783628, −17.7540950669854945603205336337, −16.82129621052041790322512918256, −14.84200070843563520275953355375, −14.15559930971820202116924778773, −12.99526678878067858682872945832, −10.155627766402356937399393510231, −9.202379299026394602511233164183, −7.55637476448840683811498847178, −6.52463798382945093665615715477, −4.352087952266413710038020469587, −1.69684863635768874722362987259, 1.89439727727827322673548076310, 3.25184704634315763579367558983, 5.32187610786013010255552184978, 8.1468873812909096439604343745, 9.25506190374578986875686392159, 10.2243066475164263214223686914, 12.098315441073938870765177762884, 13.49052825540705168042954656479, 14.59257345897278925119498398827, 16.53748812679050596166695449553, 18.1042257980689792855746520831, 19.16116341073974473193609768407, 20.49255233546053405250904959000, 21.46263588535209861066959621368, 22.215615697935760257108957604209, 24.77904073946470006018233002878, 25.53080980224011905637520309311, 26.98277429517919277521349493696, 27.88319968943100163552548930556, 29.36471755361819958419158058555, 30.229214279886677878494319982772, 31.7956751048291140967583986692, 32.19870522178916453724158234059, 34.056216121599236354662937231176, 35.65032971183591404227213235528

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