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Properties

Label 1-3751-3751.105-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.857 - 0.514i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (−0.0855 − 0.996i)2-s + (0.5 − 0.866i)3-s + (−0.985 + 0.170i)4-s + (−0.625 + 0.780i)5-s + (−0.905 − 0.424i)6-s + (−0.997 + 0.0760i)7-s + (0.254 + 0.967i)8-s + (−0.5 − 0.866i)9-s + (0.830 + 0.556i)10-s + (−0.345 + 0.938i)12-s + (0.640 + 0.768i)13-s + (0.161 + 0.986i)14-s + (0.362 + 0.931i)15-s + (0.941 − 0.336i)16-s + (−0.217 − 0.976i)17-s + (−0.820 + 0.572i)18-s + ⋯
L(s)  = 1  + (−0.0855 − 0.996i)2-s + (0.5 − 0.866i)3-s + (−0.985 + 0.170i)4-s + (−0.625 + 0.780i)5-s + (−0.905 − 0.424i)6-s + (−0.997 + 0.0760i)7-s + (0.254 + 0.967i)8-s + (−0.5 − 0.866i)9-s + (0.830 + 0.556i)10-s + (−0.345 + 0.938i)12-s + (0.640 + 0.768i)13-s + (0.161 + 0.986i)14-s + (0.362 + 0.931i)15-s + (0.941 − 0.336i)16-s + (−0.217 − 0.976i)17-s + (−0.820 + 0.572i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.857 - 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.857 - 0.514i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.857 - 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8220436562 - 0.2277700197i\)
\(L(\frac12)\) \(\approx\) \(0.8220436562 - 0.2277700197i\)
\(L(1)\) \(\approx\) \(0.6724319635 - 0.4263941310i\)
\(L(1)\) \(\approx\) \(0.6724319635 - 0.4263941310i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.0855 - 0.996i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.625 + 0.780i)T \)
7 \( 1 + (-0.997 + 0.0760i)T \)
13 \( 1 + (0.640 + 0.768i)T \)
17 \( 1 + (-0.217 - 0.976i)T \)
19 \( 1 + (0.948 + 0.318i)T \)
23 \( 1 + (-0.610 - 0.791i)T \)
29 \( 1 + (0.415 - 0.909i)T \)
37 \( 1 + (-0.640 + 0.768i)T \)
41 \( 1 + (0.290 + 0.956i)T \)
43 \( 1 + (-0.935 + 0.353i)T \)
47 \( 1 + (-0.654 + 0.755i)T \)
53 \( 1 + (-0.723 - 0.690i)T \)
59 \( 1 + (-0.290 + 0.956i)T \)
61 \( 1 + (0.974 - 0.226i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.398 - 0.917i)T \)
73 \( 1 + (-0.851 + 0.524i)T \)
79 \( 1 + (-0.888 + 0.458i)T \)
83 \( 1 + (0.879 - 0.475i)T \)
89 \( 1 + (0.870 + 0.491i)T \)
97 \( 1 + (-0.254 - 0.967i)T \)
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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

−18.74341685629933896545705134764, −17.69104717981125949904872840159, −17.13356252413702885037781878766, −16.23457595780006267935684425394, −15.94177029514008817503754729123, −15.53769061684850454871753700622, −14.82720341636347899322844839057, −13.92140391223305957935182331395, −13.30604570962630790559693074450, −12.76868159172954281919393253882, −11.876735523358554842712122221543, −10.70601886673799142622128924463, −10.16285688148269706612353076285, −9.30135818513095243138324618413, −8.8876660246264058789457765417, −8.17201629836347601120531897810, −7.59288616933092729151786852358, −6.684550820293882239917474634298, −5.64584253957148093107594115015, −5.2788540665943030605112063544, −4.30195858799394240368463528686, −3.58002568739633982900192323497, −3.26225909152292347519356417208, −1.59342179085183772354722507842, −0.347274865524725054298557213470, 0.7218657970675083479765078956, 1.7367912397981447129489374077, 2.712918097105216192448339105082, 3.08254601553688335748977417800, 3.80097972148126583351733534193, 4.63151400031322553922794776255, 5.98746004671288776531165444704, 6.5811662474008432457543971038, 7.3413961581319560405127070406, 8.13211448365600394137249674801, 8.75478256019477969056617024454, 9.66091324948656604018934968726, 10.05101971458825675958695292230, 11.19408719653935286640353560296, 11.753858822828172859946297154602, 12.13253579907742715635251735957, 13.08249117776709603152363431613, 13.62662874723667478104311347494, 14.17159174350746758871952769816, 14.85312275196730511170578218152, 15.913348921052478899439785141470, 16.42152518817475495400599099837, 17.63250033868922176209239422220, 18.23343887785427581191758622318, 18.73575121098922370139405147791

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