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Properties

Label 1-43e2-1849.130-r0-0-0
Degree $1$
Conductor $1849$
Sign $-0.854 + 0.518i$
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  + (0.391 + 0.920i)2-s + (−0.976 − 0.217i)3-s + (−0.694 + 0.719i)4-s + (0.109 + 0.994i)5-s + (−0.181 − 0.983i)6-s + (0.957 + 0.288i)7-s + (−0.934 − 0.357i)8-s + (0.905 + 0.424i)9-s + (−0.872 + 0.489i)10-s + (0.639 − 0.768i)11-s + (0.833 − 0.551i)12-s + (−0.181 + 0.983i)13-s + (0.109 + 0.994i)14-s + (0.109 − 0.994i)15-s + (−0.0365 − 0.999i)16-s + (−0.322 + 0.946i)17-s + ⋯
L(s)  = 1  + (0.391 + 0.920i)2-s + (−0.976 − 0.217i)3-s + (−0.694 + 0.719i)4-s + (0.109 + 0.994i)5-s + (−0.181 − 0.983i)6-s + (0.957 + 0.288i)7-s + (−0.934 − 0.357i)8-s + (0.905 + 0.424i)9-s + (−0.872 + 0.489i)10-s + (0.639 − 0.768i)11-s + (0.833 − 0.551i)12-s + (−0.181 + 0.983i)13-s + (0.109 + 0.994i)14-s + (0.109 − 0.994i)15-s + (−0.0365 − 0.999i)16-s + (−0.322 + 0.946i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1849\)    =    \(43^{2}\)
Sign: $-0.854 + 0.518i$
Analytic conductor: \(8.58671\)
Root analytic conductor: \(8.58671\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1849} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1849,\ (0:\ ),\ -0.854 + 0.518i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4390199040 + 1.569783601i\)
\(L(\frac12)\) \(\approx\) \(0.4390199040 + 1.569783601i\)
\(L(1)\) \(\approx\) \(0.8029347955 + 0.7639124893i\)
\(L(1)\) \(\approx\) \(0.8029347955 + 0.7639124893i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad43 \( 1 \)
good2 \( 1 + (0.391 + 0.920i)T \)
3 \( 1 + (-0.976 - 0.217i)T \)
5 \( 1 + (0.109 + 0.994i)T \)
7 \( 1 + (0.957 + 0.288i)T \)
11 \( 1 + (0.639 - 0.768i)T \)
13 \( 1 + (-0.181 + 0.983i)T \)
17 \( 1 + (-0.322 + 0.946i)T \)
19 \( 1 + T \)
23 \( 1 + (0.833 + 0.551i)T \)
29 \( 1 + (0.905 - 0.424i)T \)
31 \( 1 + (-0.181 - 0.983i)T \)
37 \( 1 + (-0.581 + 0.813i)T \)
41 \( 1 + (0.905 + 0.424i)T \)
47 \( 1 + (0.905 - 0.424i)T \)
53 \( 1 + (-0.934 + 0.357i)T \)
59 \( 1 + (0.744 - 0.667i)T \)
61 \( 1 + (0.957 + 0.288i)T \)
67 \( 1 + (0.391 + 0.920i)T \)
71 \( 1 + (0.905 + 0.424i)T \)
73 \( 1 + (-0.694 - 0.719i)T \)
79 \( 1 + (0.520 - 0.853i)T \)
83 \( 1 + (0.957 + 0.288i)T \)
89 \( 1 + (0.989 + 0.145i)T \)
97 \( 1 + (-0.181 - 0.983i)T \)
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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

−20.26768048940856793847781470000, −19.27374713070985526320246934698, −18.00060687245365797754081296170, −17.79227923527950100412941281918, −17.20300668064847510093792941797, −16.1368101164475820158158814786, −15.463677211962139129489605095200, −14.47892102814242559310761007739, −13.81550249956635537073919979381, −12.76449897701661614031741866403, −12.34744618443789094072484814835, −11.71506256959896849734730457381, −10.95089810737037627861685402438, −10.285711389576155969722503549468, −9.44130845540050729600233631172, −8.8359631547321654880133458990, −7.64720563125731484056651288358, −6.699428094712687917935197548962, −5.420758001089944928144614046580, −5.074865372239602738426734622195, −4.5280210765453655162159624411, −3.63363315217258966132653746566, −2.3122900234677772674026304969, −1.200487821347451850732087160461, −0.771191985531641369372123692401, 1.12917493906757955573034985914, 2.30886011712447842406505880259, 3.577281003340810565637552016668, 4.36032895324551837575215807429, 5.23957888514098272549262798559, 6.00479143994670418185562371513, 6.55859996541267423256511409265, 7.296317985110893565856970423864, 8.02332436588447628109322426061, 9.00935824601975040996016204203, 9.91069987721284562654264421449, 11.04932650399392023373538207516, 11.53856097520376541892143813894, 12.09301098100182970216345573811, 13.26565258248291032581245700316, 13.9103540994866204229249870025, 14.54647184180952653558623958597, 15.290232796966878570102375285912, 15.99957178987824090558399939138, 16.916357846647248465136390068154, 17.42591933989984655646535322805, 17.951756173796830256237677246077, 18.89312068527430215034139209465, 19.13561800401177724453507800783, 20.842042073178756445551437969616

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