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Properties

Label 1-1859-1859.109-r0-0-0
Degree $1$
Conductor $1859$
Sign $0.990 + 0.138i$
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  + (0.663 − 0.748i)2-s + (−0.354 + 0.935i)3-s + (−0.120 − 0.992i)4-s + (−0.239 + 0.970i)5-s + (0.464 + 0.885i)6-s + (−0.822 − 0.568i)7-s + (−0.822 − 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (0.970 + 0.239i)12-s + (−0.970 + 0.239i)14-s + (−0.822 − 0.568i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (−0.992 + 0.120i)18-s + i·19-s + ⋯
L(s)  = 1  + (0.663 − 0.748i)2-s + (−0.354 + 0.935i)3-s + (−0.120 − 0.992i)4-s + (−0.239 + 0.970i)5-s + (0.464 + 0.885i)6-s + (−0.822 − 0.568i)7-s + (−0.822 − 0.568i)8-s + (−0.748 − 0.663i)9-s + (0.568 + 0.822i)10-s + (0.970 + 0.239i)12-s + (−0.970 + 0.239i)14-s + (−0.822 − 0.568i)15-s + (−0.970 + 0.239i)16-s + (0.568 − 0.822i)17-s + (−0.992 + 0.120i)18-s + i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.138i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $0.990 + 0.138i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ 0.990 + 0.138i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.289185452 + 0.09001483940i\)
\(L(\frac12)\) \(\approx\) \(1.289185452 + 0.09001483940i\)
\(L(1)\) \(\approx\) \(1.035178820 - 0.09449591661i\)
\(L(1)\) \(\approx\) \(1.035178820 - 0.09449591661i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.663 - 0.748i)T \)
3 \( 1 + (-0.354 + 0.935i)T \)
5 \( 1 + (-0.239 + 0.970i)T \)
7 \( 1 + (-0.822 - 0.568i)T \)
17 \( 1 + (0.568 - 0.822i)T \)
19 \( 1 + iT \)
23 \( 1 - T \)
29 \( 1 + (0.748 + 0.663i)T \)
31 \( 1 + (-0.464 - 0.885i)T \)
37 \( 1 + (0.464 + 0.885i)T \)
41 \( 1 + (-0.935 - 0.354i)T \)
43 \( 1 + (0.885 + 0.464i)T \)
47 \( 1 + (0.992 + 0.120i)T \)
53 \( 1 + (0.568 - 0.822i)T \)
59 \( 1 + (0.239 - 0.970i)T \)
61 \( 1 + (-0.568 - 0.822i)T \)
67 \( 1 + (-0.992 - 0.120i)T \)
71 \( 1 + (0.935 + 0.354i)T \)
73 \( 1 + (0.663 + 0.748i)T \)
79 \( 1 + (-0.120 + 0.992i)T \)
83 \( 1 + (0.935 - 0.354i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.239 + 0.970i)T \)
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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

−19.82292155672418027940402144739, −19.57568607099660872237968686173, −18.46047777519318858788015280542, −17.762090306814417349667283338838, −17.01737491884481951947940285869, −16.42935058108454067970946563076, −15.779445198030957324923735160958, −15.06336156321931392373082243163, −13.95103002116006190272413786956, −13.39196674707731049367104502795, −12.67896954250580981235866740972, −12.19482375318320145972872920564, −11.72399607317290148918567289756, −10.4536827193178125466577428970, −9.11759995464028321611407789749, −8.64333650668466808890460689580, −7.78864614542081134549177280605, −7.11966486495893914680337273257, −6.08451941747611648135813795658, −5.7800532629238354964677350955, −4.86025777487205201468415928313, −3.938928484813290309305250498448, −2.902864802209768443854378728249, −1.96473910273712175785793101821, −0.559528495858004020117945485252, 0.72688415953746207478566569798, 2.28971721071962701830390344391, 3.230788914196797817684645386628, 3.66295306903602731753629753907, 4.4038900798190977413582705103, 5.45857173512813882517732019759, 6.18354747108491519913541913016, 6.85498270988760908042894130633, 8.00904743764390394041598931345, 9.38056739213593623692458322333, 9.93180529665575039501818560312, 10.41271924754740335397538776954, 11.1204616982263191681941958030, 11.8921878042196528462910488459, 12.471191276431125115039367346417, 13.68427729872532494962053133602, 14.1943702447341938542575321767, 14.84168069455716377327852336006, 15.71562417765408724154738076499, 16.19586989714972969454328745348, 17.114534210528998999104941389766, 18.24283957111288325742251024399, 18.70741034225126905195789166859, 19.63930221402816424521192355867, 20.26399107996336643596387400543

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