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Properties

Label 1-3893-3893.1504-r0-0-0
Degree $1$
Conductor $3893$
Sign $0.997 + 0.0736i$
Analytic cond. $18.0790$
Root an. cond. $18.0790$
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  + (−0.996 + 0.0825i)2-s + (0.885 − 0.463i)3-s + (0.986 − 0.164i)4-s + (−0.921 − 0.389i)5-s + (−0.844 + 0.535i)6-s + (−0.687 − 0.726i)7-s + (−0.969 + 0.245i)8-s + (0.569 − 0.821i)9-s + (0.950 + 0.311i)10-s + (0.931 + 0.363i)11-s + (0.797 − 0.603i)12-s + (−0.789 − 0.614i)13-s + (0.744 + 0.667i)14-s + (−0.996 + 0.0825i)15-s + (0.945 − 0.324i)16-s + ⋯
L(s)  = 1  + (−0.996 + 0.0825i)2-s + (0.885 − 0.463i)3-s + (0.986 − 0.164i)4-s + (−0.921 − 0.389i)5-s + (−0.844 + 0.535i)6-s + (−0.687 − 0.726i)7-s + (−0.969 + 0.245i)8-s + (0.569 − 0.821i)9-s + (0.950 + 0.311i)10-s + (0.931 + 0.363i)11-s + (0.797 − 0.603i)12-s + (−0.789 − 0.614i)13-s + (0.744 + 0.667i)14-s + (−0.996 + 0.0825i)15-s + (0.945 − 0.324i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3893 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0736i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3893 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0736i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3893\)    =    \(17 \cdot 229\)
Sign: $0.997 + 0.0736i$
Analytic conductor: \(18.0790\)
Root analytic conductor: \(18.0790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3893} (1504, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3893,\ (0:\ ),\ 0.997 + 0.0736i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.052211485 + 0.03881660831i\)
\(L(\frac12)\) \(\approx\) \(1.052211485 + 0.03881660831i\)
\(L(1)\) \(\approx\) \(0.7699887714 - 0.1527965092i\)
\(L(1)\) \(\approx\) \(0.7699887714 - 0.1527965092i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
229 \( 1 \)
good2 \( 1 + (-0.996 + 0.0825i)T \)
3 \( 1 + (0.885 - 0.463i)T \)
5 \( 1 + (-0.921 - 0.389i)T \)
7 \( 1 + (-0.687 - 0.726i)T \)
11 \( 1 + (0.931 + 0.363i)T \)
13 \( 1 + (-0.789 - 0.614i)T \)
19 \( 1 + (0.990 - 0.137i)T \)
23 \( 1 + (0.311 + 0.950i)T \)
29 \( 1 + (0.232 + 0.972i)T \)
31 \( 1 + (-0.625 - 0.780i)T \)
37 \( 1 + (-0.558 + 0.829i)T \)
41 \( 1 + (0.941 - 0.337i)T \)
43 \( 1 + (-0.324 + 0.945i)T \)
47 \( 1 + (0.821 + 0.569i)T \)
53 \( 1 + (-0.735 + 0.677i)T \)
59 \( 1 + (-0.981 - 0.191i)T \)
61 \( 1 + (0.646 + 0.763i)T \)
67 \( 1 + (0.904 - 0.426i)T \)
71 \( 1 + (-0.150 + 0.988i)T \)
73 \( 1 + (0.999 + 0.0137i)T \)
79 \( 1 + (-0.232 + 0.972i)T \)
83 \( 1 + (-0.656 + 0.754i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.872 - 0.488i)T \)
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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

−18.78535749157132835107833178248, −18.12907018670551811802976850704, −16.96426130781790348415859099476, −16.43315015470077413882012037280, −15.78123872370182576590758945962, −15.38905857548402559096556713705, −14.47197958551407936152657546829, −14.18540409717234895066790011302, −12.7984537094148129638954965747, −12.1378125320148372473965312391, −11.59720725008853549004936611041, −10.77940484577065870930182930050, −10.02189781505284256828530676497, −9.33528554436332732059568101558, −8.88035644561226274069417451797, −8.242643595703527441456961477877, −7.356495654707816958965230485082, −6.91957543127473230868620497434, −6.049071614982942385226908541, −4.873205434049140853567247261273, −3.81405323108911692209971961616, −3.303021234399222096450834576165, −2.56910782696887031876741345628, −1.83850550374828820004559428472, −0.46414218900340638178735194531, 0.91672196881401417084601296330, 1.30820772402148474321726013690, 2.58593104305287077701470046765, 3.32101931418222725855403171581, 3.86843235914855117127205445531, 4.99364844993964667039071470382, 6.17472243654880551508732716970, 7.12536342123038218435054580834, 7.334406129190852689658916210368, 7.94133495841148148378242610049, 8.82138483217245550906501852690, 9.506485964052058341976311103694, 9.80123556779922533546218465516, 10.910626485780589627475958063347, 11.635606950004721115812635108047, 12.45683951676992623284711011038, 12.77090564651902668149083346613, 13.90906119828963626660729949026, 14.546034115962049099099068299964, 15.38043056160729558396832649142, 15.72453263442844410069150926712, 16.63668103675659631495667143144, 17.180480071301975584419791152433, 17.893301603421180571663707222104, 18.724999358625065480875206578935

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