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Properties

Label 1-1859-1859.433-r0-0-0
Degree $1$
Conductor $1859$
Sign $0.506 + 0.862i$
Analytic cond. $8.63315$
Root an. cond. $8.63315$
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  + (−0.457 − 0.889i)2-s + (−0.877 + 0.478i)3-s + (−0.581 + 0.813i)4-s + (0.906 + 0.421i)5-s + (0.827 + 0.561i)6-s + (0.554 + 0.832i)7-s + (0.989 + 0.144i)8-s + (0.541 − 0.840i)9-s + (−0.0402 − 0.999i)10-s + (0.120 − 0.992i)12-s + (0.485 − 0.873i)14-s + (−0.997 + 0.0643i)15-s + (−0.324 − 0.945i)16-s + (0.937 + 0.347i)17-s + (−0.995 − 0.0965i)18-s + (−0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (−0.457 − 0.889i)2-s + (−0.877 + 0.478i)3-s + (−0.581 + 0.813i)4-s + (0.906 + 0.421i)5-s + (0.827 + 0.561i)6-s + (0.554 + 0.832i)7-s + (0.989 + 0.144i)8-s + (0.541 − 0.840i)9-s + (−0.0402 − 0.999i)10-s + (0.120 − 0.992i)12-s + (0.485 − 0.873i)14-s + (−0.997 + 0.0643i)15-s + (−0.324 − 0.945i)16-s + (0.937 + 0.347i)17-s + (−0.995 − 0.0965i)18-s + (−0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.506 + 0.862i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ 0.506 + 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9296512973 + 0.5322762420i\)
\(L(\frac12)\) \(\approx\) \(0.9296512973 + 0.5322762420i\)
\(L(1)\) \(\approx\) \(0.7970811137 + 0.04913386897i\)
\(L(1)\) \(\approx\) \(0.7970811137 + 0.04913386897i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.457 - 0.889i)T \)
3 \( 1 + (-0.877 + 0.478i)T \)
5 \( 1 + (0.906 + 0.421i)T \)
7 \( 1 + (0.554 + 0.832i)T \)
17 \( 1 + (0.937 + 0.347i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.704 + 0.709i)T \)
31 \( 1 + (0.527 + 0.849i)T \)
37 \( 1 + (0.984 + 0.176i)T \)
41 \( 1 + (0.877 - 0.478i)T \)
43 \( 1 + (0.692 - 0.721i)T \)
47 \( 1 + (-0.995 + 0.0965i)T \)
53 \( 1 + (0.715 + 0.698i)T \)
59 \( 1 + (0.999 + 0.0161i)T \)
61 \( 1 + (0.247 + 0.968i)T \)
67 \( 1 + (-0.948 - 0.316i)T \)
71 \( 1 + (0.231 - 0.972i)T \)
73 \( 1 + (-0.836 - 0.548i)T \)
79 \( 1 + (-0.861 - 0.506i)T \)
83 \( 1 + (0.607 + 0.794i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (-0.0884 + 0.996i)T \)
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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

−19.75498625608936922358499243815, −19.04408325814321591707189970233, −18.167279225076053902412777664683, −17.63648206548253310995076853563, −17.12729851188184006513499647924, −16.58775986546478956750003772091, −15.94482055932190026710644442753, −14.755690325574269680815021020587, −14.153497796151933170289051108675, −13.21012550231606771663609052771, −13.029990121028791723348117625310, −11.61325619873078845224089424706, −10.98606559883892401507939330506, −9.95226883208398722295053146506, −9.71693311051952032201239140467, −8.413899844754327746780525943020, −7.73004649899383412651109899195, −7.074466328102331680275097961825, −6.09968643475499238427503139520, −5.69836436538472167949480654568, −4.781333450039395460727503889001, −4.16521943772939237748855988934, −2.218714313170056600100521926458, −1.34765680350084267152798021921, −0.59222382480436424562425420245, 1.12093348958509202046079689921, 1.97424787484398393804429131969, 2.833019687456056711557608057282, 3.88461951051134456151834080599, 4.78189095805119727153691610309, 5.61152283129246703412288287145, 6.25871948585813163968817565618, 7.38034789591261953853869140550, 8.46496710277049254900486407353, 9.15302949099893838504769127775, 9.951530155301439777479670555175, 10.583082715760328789205784272775, 11.04060213910273553200777502620, 12.1165335529267136550960908266, 12.40407148968809414167302138101, 13.34580923769834615523448053950, 14.504323901983600012823225337459, 14.86455336156097849667900115483, 16.21239635146933132606568612767, 16.73180612648983951006630246041, 17.57794447946232835140966979935, 18.00769787501277997728689463680, 18.64072478170107072100550188958, 19.280965837026275808795212005208, 20.62163199687811089073298547885

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