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Properties

Label 1-43e2-1849.302-r0-0-0
Degree $1$
Conductor $1849$
Sign $0.546 + 0.837i$
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  + (0.109 − 0.994i)2-s + (−0.872 − 0.489i)3-s + (−0.976 − 0.217i)4-s + (0.252 + 0.967i)5-s + (−0.581 + 0.813i)6-s + (−0.934 + 0.357i)7-s + (−0.322 + 0.946i)8-s + (0.520 + 0.853i)9-s + (0.989 − 0.145i)10-s + (−0.457 − 0.889i)11-s + (0.744 + 0.667i)12-s + (−0.581 − 0.813i)13-s + (0.252 + 0.967i)14-s + (0.252 − 0.967i)15-s + (0.905 + 0.424i)16-s + (−0.694 + 0.719i)17-s + ⋯
L(s)  = 1  + (0.109 − 0.994i)2-s + (−0.872 − 0.489i)3-s + (−0.976 − 0.217i)4-s + (0.252 + 0.967i)5-s + (−0.581 + 0.813i)6-s + (−0.934 + 0.357i)7-s + (−0.322 + 0.946i)8-s + (0.520 + 0.853i)9-s + (0.989 − 0.145i)10-s + (−0.457 − 0.889i)11-s + (0.744 + 0.667i)12-s + (−0.581 − 0.813i)13-s + (0.252 + 0.967i)14-s + (0.252 − 0.967i)15-s + (0.905 + 0.424i)16-s + (−0.694 + 0.719i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 + 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $0.546 + 0.837i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (302, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ 0.546 + 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2577765223 + 0.1395314819i\)
\(L(\frac12)\) \(\approx\) \(0.2577765223 + 0.1395314819i\)
\(L(1)\) \(\approx\) \(0.5418917041 - 0.2589506454i\)
\(L(1)\) \(\approx\) \(0.5418917041 - 0.2589506454i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.109 - 0.994i)T \)
3 \( 1 + (-0.872 - 0.489i)T \)
5 \( 1 + (0.252 + 0.967i)T \)
7 \( 1 + (-0.934 + 0.357i)T \)
11 \( 1 + (-0.457 - 0.889i)T \)
13 \( 1 + (-0.581 - 0.813i)T \)
17 \( 1 + (-0.694 + 0.719i)T \)
19 \( 1 + T \)
23 \( 1 + (0.744 - 0.667i)T \)
29 \( 1 + (0.520 - 0.853i)T \)
31 \( 1 + (-0.581 + 0.813i)T \)
37 \( 1 + (0.391 - 0.920i)T \)
41 \( 1 + (0.520 + 0.853i)T \)
47 \( 1 + (0.520 - 0.853i)T \)
53 \( 1 + (-0.322 - 0.946i)T \)
59 \( 1 + (-0.791 + 0.611i)T \)
61 \( 1 + (-0.934 + 0.357i)T \)
67 \( 1 + (0.109 - 0.994i)T \)
71 \( 1 + (0.520 + 0.853i)T \)
73 \( 1 + (-0.976 + 0.217i)T \)
79 \( 1 + (0.957 - 0.288i)T \)
83 \( 1 + (-0.934 + 0.357i)T \)
89 \( 1 + (-0.181 - 0.983i)T \)
97 \( 1 + (-0.581 + 0.813i)T \)
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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

−20.2117953335127384697782708502, −19.12351002160825721846199625490, −18.17639735218018395072847981301, −17.56292231633807620818890490375, −16.86418888738385227621523397139, −16.43080537774495444725300417825, −15.71491050518089153609382398741, −15.28907876841364462480110579492, −14.04449415155923185814538769786, −13.39704307883182349872293194295, −12.557894509233820404664023274174, −12.18122857114080866048685651053, −11.01252662089330517568444476535, −9.75397060586601372749783950656, −9.57206660788960478427107521660, −8.93566272170366516124971240588, −7.46689767820485717753454729873, −7.0706902267974669086318959562, −6.14703961162534477319685620293, −5.35317059272400183881525804508, −4.68538700603148607870534031267, −4.20351886888338278701494734864, −2.99662763157078519465397387102, −1.31158229942210802567856931359, −0.14893024618990139069534918516, 0.91980559117276297873224186171, 2.2710940518707449318793768178, 2.828659114401046165547299119270, 3.629957201959197609429512524950, 4.89526714657392469653613610080, 5.733957379304427750087396684875, 6.21812141305236975998577430550, 7.21578433991610330693227632176, 8.14244277917507997030051986850, 9.21606520319938634738528372795, 10.11908756755769106993440482311, 10.616550498123438881277615742835, 11.21136092360224109462367398964, 12.04011229204262564691965028430, 12.81468212904688442760388851611, 13.279603481753576497085802388237, 14.03937052327847912768722321629, 15.02747402428235657748916473412, 15.84968288828699171899734333646, 16.79852196154032200557295047037, 17.65396292907929848841817018417, 18.227840131873515877066513873631, 18.738748798987218929413136131346, 19.47384978845614324806433704146, 19.92690862202988799020888671816

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