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Properties

Label 1-43e2-1849.9-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.999 + 0.00339i$
Analytic cond. $8.58671$
Root an. cond. $8.58671$
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  + (−0.283 − 0.959i)2-s + (0.999 + 0.00695i)3-s + (−0.839 + 0.543i)4-s + (0.225 + 0.974i)5-s + (−0.276 − 0.961i)6-s + (−0.133 + 0.991i)7-s + (0.758 + 0.651i)8-s + (0.999 + 0.0139i)9-s + (0.870 − 0.492i)10-s + (−0.751 − 0.659i)11-s + (−0.843 + 0.537i)12-s + (0.579 + 0.815i)13-s + (0.988 − 0.152i)14-s + (0.219 + 0.975i)15-s + (0.410 − 0.911i)16-s + (−0.0643 − 0.997i)17-s + ⋯
L(s)  = 1  + (−0.283 − 0.959i)2-s + (0.999 + 0.00695i)3-s + (−0.839 + 0.543i)4-s + (0.225 + 0.974i)5-s + (−0.276 − 0.961i)6-s + (−0.133 + 0.991i)7-s + (0.758 + 0.651i)8-s + (0.999 + 0.0139i)9-s + (0.870 − 0.492i)10-s + (−0.751 − 0.659i)11-s + (−0.843 + 0.537i)12-s + (0.579 + 0.815i)13-s + (0.988 − 0.152i)14-s + (0.219 + 0.975i)15-s + (0.410 − 0.911i)16-s + (−0.0643 − 0.997i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00339i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.999 + 0.00339i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.999 + 0.00339i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.105663636 + 0.003577687277i\)
\(L(\frac12)\) \(\approx\) \(2.105663636 + 0.003577687277i\)
\(L(1)\) \(\approx\) \(1.329860307 - 0.1773941718i\)
\(L(1)\) \(\approx\) \(1.329860307 - 0.1773941718i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (-0.283 - 0.959i)T \)
3 \( 1 + (0.999 + 0.00695i)T \)
5 \( 1 + (0.225 + 0.974i)T \)
7 \( 1 + (-0.133 + 0.991i)T \)
11 \( 1 + (-0.751 - 0.659i)T \)
13 \( 1 + (0.579 + 0.815i)T \)
17 \( 1 + (-0.0643 - 0.997i)T \)
19 \( 1 + (0.826 - 0.563i)T \)
23 \( 1 + (0.983 - 0.179i)T \)
29 \( 1 + (0.959 + 0.281i)T \)
31 \( 1 + (0.998 - 0.0556i)T \)
37 \( 1 + (0.676 - 0.736i)T \)
41 \( 1 + (0.352 + 0.935i)T \)
47 \( 1 + (-0.512 - 0.858i)T \)
53 \( 1 + (0.185 + 0.982i)T \)
59 \( 1 + (-0.678 - 0.734i)T \)
61 \( 1 + (0.995 - 0.0903i)T \)
67 \( 1 + (-0.444 + 0.895i)T \)
71 \( 1 + (0.612 + 0.790i)T \)
73 \( 1 + (-0.387 - 0.921i)T \)
79 \( 1 + (-0.950 - 0.311i)T \)
83 \( 1 + (-0.576 - 0.817i)T \)
89 \( 1 + (-0.587 + 0.809i)T \)
97 \( 1 + (0.312 - 0.949i)T \)
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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

−20.003749193363896647505491173324, −19.521182516304888640582489868222, −18.60203187792706819215739959818, −17.7126507610445743605683098690, −17.262340308383523638615383209780, −16.308993178399889809478128633687, −15.726016916676596590788927544429, −15.13954737402005160413277496432, −14.22000065351112855373598197441, −13.48472140112050930272767413032, −13.09722182289388780072342018240, −12.42546437110435921057368882098, −10.71674324949208252141556156643, −10.00628977871133331381261656754, −9.58904032153556628175069245064, −8.45471252304167270956990331734, −8.11978679936647108144972818883, −7.438372365511371216246508985950, −6.53210400530973627103051170975, −5.520833818061369160504901535377, −4.6480652999316771484676571065, −4.04172922236096815373305953531, −2.9994447167864038297323442806, −1.525826107770986445788993463060, −0.904320959754901080277327752609, 1.09321424736415556826234979357, 2.2736635175283613468420886529, 2.87379558136618856690924339689, 3.17123448197828499258817212108, 4.405465527248314183198172035359, 5.32085244906954025873132709713, 6.53083557926904116435047214523, 7.39933017609017660124616834581, 8.28301035146393936073263539955, 8.983705952353121259078363424412, 9.54935024335934252022996614847, 10.30164237978347629780352982848, 11.24060864818422223503876125409, 11.69492466132086326102249124194, 12.8188169943653558404844108435, 13.53355610479355517821470790713, 13.99790084377024474859658377295, 14.7754151895878009843666490400, 15.74406645427991906320235048376, 16.25182634935569047457245144747, 17.70758024723537189015190471680, 18.35033261091849521901081168147, 18.71939150403829965954100507621, 19.247946948388983384714698621983, 20.06942435918784183705270978906

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