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Properties

Label 1-43e2-1849.1764-r0-0-0
Degree $1$
Conductor $1849$
Sign $-0.251 + 0.967i$
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  + (0.252 + 0.967i)2-s + (0.989 − 0.145i)3-s + (−0.872 + 0.489i)4-s + (−0.997 − 0.0729i)5-s + (0.391 + 0.920i)6-s + (−0.322 + 0.946i)7-s + (−0.694 − 0.719i)8-s + (0.957 − 0.288i)9-s + (−0.181 − 0.983i)10-s + (0.833 + 0.551i)11-s + (−0.791 + 0.611i)12-s + (0.391 − 0.920i)13-s + (−0.997 − 0.0729i)14-s + (−0.997 + 0.0729i)15-s + (0.520 − 0.853i)16-s + (−0.976 + 0.217i)17-s + ⋯
L(s)  = 1  + (0.252 + 0.967i)2-s + (0.989 − 0.145i)3-s + (−0.872 + 0.489i)4-s + (−0.997 − 0.0729i)5-s + (0.391 + 0.920i)6-s + (−0.322 + 0.946i)7-s + (−0.694 − 0.719i)8-s + (0.957 − 0.288i)9-s + (−0.181 − 0.983i)10-s + (0.833 + 0.551i)11-s + (−0.791 + 0.611i)12-s + (0.391 − 0.920i)13-s + (−0.997 − 0.0729i)14-s + (−0.997 + 0.0729i)15-s + (0.520 − 0.853i)16-s + (−0.976 + 0.217i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.251 + 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-0.251 + 0.967i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (1764, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ -0.251 + 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.238389463 + 1.600954696i\)
\(L(\frac12)\) \(\approx\) \(1.238389463 + 1.600954696i\)
\(L(1)\) \(\approx\) \(1.144077943 + 0.7362061853i\)
\(L(1)\) \(\approx\) \(1.144077943 + 0.7362061853i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.252 + 0.967i)T \)
3 \( 1 + (0.989 - 0.145i)T \)
5 \( 1 + (-0.997 - 0.0729i)T \)
7 \( 1 + (-0.322 + 0.946i)T \)
11 \( 1 + (0.833 + 0.551i)T \)
13 \( 1 + (0.391 - 0.920i)T \)
17 \( 1 + (-0.976 + 0.217i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.791 - 0.611i)T \)
29 \( 1 + (0.957 + 0.288i)T \)
31 \( 1 + (0.391 + 0.920i)T \)
37 \( 1 + (0.109 - 0.994i)T \)
41 \( 1 + (0.957 - 0.288i)T \)
47 \( 1 + (0.957 + 0.288i)T \)
53 \( 1 + (-0.694 + 0.719i)T \)
59 \( 1 + (-0.0365 - 0.999i)T \)
61 \( 1 + (-0.322 + 0.946i)T \)
67 \( 1 + (0.252 + 0.967i)T \)
71 \( 1 + (0.957 - 0.288i)T \)
73 \( 1 + (-0.872 - 0.489i)T \)
79 \( 1 + (-0.934 + 0.357i)T \)
83 \( 1 + (-0.322 + 0.946i)T \)
89 \( 1 + (-0.581 - 0.813i)T \)
97 \( 1 + (0.391 + 0.920i)T \)
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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

−20.05007619590625715120850386921, −19.3742504932578844776736576465, −18.90706752741521190602584150020, −18.05787191447730116917514186726, −16.958368618476164762960871082991, −16.029034297401767155216567025508, −15.47158072773484763012265005548, −14.42311786925383038537454752215, −13.83627040267470414790570694408, −13.50496231365672291808547341766, −12.45581315278851152901932139861, −11.532129571264553890239856880217, −11.18385692856068085867515932829, −10.092847373881750701987618820184, −9.46968301557376657240299732821, −8.71928724566784938590770023631, −7.98270323960089095641774884538, −7.071717450658996498778540261801, −6.19715326968094208745789214135, −4.54467995365895308443223844802, −4.21826916700085665889812993561, −3.51666514924941781557379940605, −2.859046861604984767794233135420, −1.65736355522438792398137626827, −0.75245684391407212713128456034, 0.989210257875474040854785162850, 2.52139140808455751554948220116, 3.30890769399549845851362677630, 4.07724446603241294695292209417, 4.77963809366699014685799388150, 5.9399454372471334718621160621, 6.75393287105709201996410999508, 7.46482656346706760985267846379, 8.238789581813182651155949054943, 8.80566506240367086010837836787, 9.35592044583863197504416977121, 10.41253092237583293812661353184, 11.77973619826408858597372327881, 12.50345515676252767155714816955, 12.83591627613279566002368107096, 14.0292967182687123094153582137, 14.47665338799436334122784094883, 15.39159080804207171455992471768, 15.71309267794149073095167536492, 16.18479784938117507441353510041, 17.57558217260182735005193702647, 18.09451196129820864809331503207, 18.82525521230631409947113391748, 19.70699924046865133429612058081, 20.087417163506656207938629652967

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