Properties

Label 1-43e2-1849.4-r0-0-0
Degree $1$
Conductor $1849$
Sign $-0.855 + 0.517i$
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  + (0.944 − 0.327i)2-s + (−0.283 − 0.959i)3-s + (0.785 − 0.619i)4-s + (0.914 − 0.405i)5-s + (−0.581 − 0.813i)6-s + (−0.934 − 0.357i)7-s + (0.538 − 0.842i)8-s + (−0.839 + 0.543i)9-s + (0.730 − 0.682i)10-s + (0.797 − 0.602i)11-s + (−0.816 − 0.577i)12-s + (−0.998 + 0.0521i)13-s + (−0.999 − 0.0313i)14-s + (−0.647 − 0.761i)15-s + (0.232 − 0.972i)16-s + (−0.995 + 0.0937i)17-s + ⋯
L(s)  = 1  + (0.944 − 0.327i)2-s + (−0.283 − 0.959i)3-s + (0.785 − 0.619i)4-s + (0.914 − 0.405i)5-s + (−0.581 − 0.813i)6-s + (−0.934 − 0.357i)7-s + (0.538 − 0.842i)8-s + (−0.839 + 0.543i)9-s + (0.730 − 0.682i)10-s + (0.797 − 0.602i)11-s + (−0.816 − 0.577i)12-s + (−0.998 + 0.0521i)13-s + (−0.999 − 0.0313i)14-s + (−0.647 − 0.761i)15-s + (0.232 − 0.972i)16-s + (−0.995 + 0.0937i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.855 + 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-0.855 + 0.517i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ -0.855 + 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4889393010 - 1.754844974i\)
\(L(\frac12)\) \(\approx\) \(-0.4889393010 - 1.754844974i\)
\(L(1)\) \(\approx\) \(1.023175819 - 1.098834765i\)
\(L(1)\) \(\approx\) \(1.023175819 - 1.098834765i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.944 - 0.327i)T \)
3 \( 1 + (-0.283 - 0.959i)T \)
5 \( 1 + (0.914 - 0.405i)T \)
7 \( 1 + (-0.934 - 0.357i)T \)
11 \( 1 + (0.797 - 0.602i)T \)
13 \( 1 + (-0.998 + 0.0521i)T \)
17 \( 1 + (-0.995 + 0.0937i)T \)
19 \( 1 + (-0.900 + 0.433i)T \)
23 \( 1 + (-0.381 - 0.924i)T \)
29 \( 1 + (0.716 - 0.697i)T \)
31 \( 1 + (-0.663 + 0.748i)T \)
37 \( 1 + (0.391 + 0.920i)T \)
41 \( 1 + (-0.948 - 0.317i)T \)
47 \( 1 + (-0.839 - 0.543i)T \)
53 \( 1 + (0.538 + 0.842i)T \)
59 \( 1 + (-0.971 + 0.237i)T \)
61 \( 1 + (0.556 - 0.831i)T \)
67 \( 1 + (-0.529 - 0.848i)T \)
71 \( 1 + (-0.0989 + 0.995i)T \)
73 \( 1 + (-0.438 - 0.898i)T \)
79 \( 1 + (0.957 + 0.288i)T \)
83 \( 1 + (0.556 - 0.831i)T \)
89 \( 1 + (-0.918 - 0.395i)T \)
97 \( 1 + (0.171 + 0.985i)T \)
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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

−20.85729520771191554252606966436, −19.81854237625068497580055005396, −19.58957145702067807067488142505, −17.958586693205537926885935421413, −17.432576799016588836881363958875, −16.750901000484071669721233638123, −16.10247929817292392614215902426, −15.08831416097158220443282675589, −14.9346810180754046952937969325, −14.04616066350223682277056717985, −13.186573216795557398052573426705, −12.51164055106212431286919240484, −11.68127971951887605363997741459, −10.91753980811717053217583744265, −10.030998250810499441110880792454, −9.43504126445395274863066331783, −8.69988254449782542939345403782, −7.18966281710958158937512898769, −6.57853124465290314065193502175, −5.97306468788027618553204151461, −5.15161189087843953278375067207, −4.40574846988104440179667815362, −3.5506933678519777306401440452, −2.673481044330457752470539127253, −1.989441096979453830193462967560, 0.40795649415746229462140725054, 1.55688657090131282946444407910, 2.28321691356685051232149557197, 3.0773248091837640121239100267, 4.259814223601324377229344448544, 5.04693242573748769546851146371, 6.13646278001381922109861416168, 6.406103308886987572400879697661, 7.021682937317368911922115981135, 8.3150514191849393701519805973, 9.22820376135792330298739648790, 10.19064154838072275968110147209, 10.760480453976851899085331925, 11.891644121971524816884581951413, 12.37374550690678435262450797957, 13.06002363360038599700387739912, 13.62387092605756538796413920786, 14.18752441717815209738179754317, 14.982525218148808451734023553617, 16.217663641260401159637218804523, 16.81590229337912114348656997608, 17.28816916169079150070625665388, 18.4247739683525866661574844780, 19.149401473313665567018636962122, 19.86739516856877736503296864222

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