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Properties

Label 1-1859-1859.542-r0-0-0
Degree $1$
Conductor $1859$
Sign $-0.952 - 0.305i$
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  + (−0.962 − 0.270i)2-s + (−0.789 − 0.613i)3-s + (0.853 + 0.520i)4-s + (0.779 − 0.626i)5-s + (0.594 + 0.804i)6-s + (0.513 + 0.857i)7-s + (−0.681 − 0.732i)8-s + (0.247 + 0.968i)9-s + (−0.919 + 0.391i)10-s + (−0.354 − 0.935i)12-s + (−0.262 − 0.964i)14-s + (−0.999 + 0.0161i)15-s + (0.457 + 0.889i)16-s + (0.0884 − 0.996i)17-s + (0.0241 − 0.999i)18-s + (−0.104 − 0.994i)19-s + ⋯
L(s)  = 1  + (−0.962 − 0.270i)2-s + (−0.789 − 0.613i)3-s + (0.853 + 0.520i)4-s + (0.779 − 0.626i)5-s + (0.594 + 0.804i)6-s + (0.513 + 0.857i)7-s + (−0.681 − 0.732i)8-s + (0.247 + 0.968i)9-s + (−0.919 + 0.391i)10-s + (−0.354 − 0.935i)12-s + (−0.262 − 0.964i)14-s + (−0.999 + 0.0161i)15-s + (0.457 + 0.889i)16-s + (0.0884 − 0.996i)17-s + (0.0241 − 0.999i)18-s + (−0.104 − 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 - 0.305i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-0.952 - 0.305i$
Analytic conductor: \(8.63315\)
Root analytic conductor: \(8.63315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1859} (542, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1859,\ (0:\ ),\ -0.952 - 0.305i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08732495110 - 0.5573446919i\)
\(L(\frac12)\) \(\approx\) \(0.08732495110 - 0.5573446919i\)
\(L(1)\) \(\approx\) \(0.5413537374 - 0.2604819843i\)
\(L(1)\) \(\approx\) \(0.5413537374 - 0.2604819843i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.962 - 0.270i)T \)
3 \( 1 + (-0.789 - 0.613i)T \)
5 \( 1 + (0.779 - 0.626i)T \)
7 \( 1 + (0.513 + 0.857i)T \)
17 \( 1 + (0.0884 - 0.996i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (-0.554 + 0.832i)T \)
31 \( 1 + (-0.861 + 0.506i)T \)
37 \( 1 + (0.737 - 0.675i)T \)
41 \( 1 + (-0.789 - 0.613i)T \)
43 \( 1 + (-0.200 - 0.979i)T \)
47 \( 1 + (0.0241 + 0.999i)T \)
53 \( 1 + (0.981 + 0.192i)T \)
59 \( 1 + (-0.704 - 0.709i)T \)
61 \( 1 + (-0.324 + 0.945i)T \)
67 \( 1 + (-0.996 - 0.0804i)T \)
71 \( 1 + (0.899 + 0.435i)T \)
73 \( 1 + (-0.989 - 0.144i)T \)
79 \( 1 + (-0.607 - 0.794i)T \)
83 \( 1 + (-0.527 - 0.849i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.932 + 0.362i)T \)
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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

−20.44078297621981796847598841097, −19.71767208387944234515774909825, −18.60954898153585470194535315437, −18.14424361778151049519788385713, −17.436912723765693657965553446476, −16.78667413541061210855407064516, −16.552574319236049977040531432092, −15.15880926377972684191197559655, −14.96266985661277572385022372614, −14.03071097145651726341455472259, −13.06485234560034253832078351862, −11.84369038445739746548854182430, −11.20385514664691514230809525309, −10.57554707080731872099729926315, −9.952334403631015276824851612369, −9.54844502220220212836010872601, −8.31601744213022449881184100586, −7.55112919586945131901665760616, −6.685578131475907915894572723909, −5.96509260517365371453483128883, −5.43748090016761945211298859573, −4.19353345911921303670533564454, −3.31771521559189312255640130810, −1.9046683361958919598844654730, −1.26343107422113655477308515366, 0.309877049942714441469683756179, 1.40358604030815639869896094448, 2.09602861656598059784331677981, 2.81882480496747198344522668335, 4.52403326887448466833294676353, 5.37508984971584652722804431922, 6.00435893822502790052561533341, 6.94186450725957783284462266961, 7.6164133450762012861070111742, 8.75204699690803537601323880118, 9.002026164791684060542214150286, 10.070487419874594178173626187598, 10.84939054882081147461325323483, 11.55073120932775788880011410294, 12.27897488324471706112160306728, 12.76361790157298650958691340730, 13.665144094816506874561715433, 14.66716226253232626846696504960, 15.825450011155712888815323128050, 16.32372273580182521867455142781, 17.09236152784068107464237257737, 17.70916465942985948036737792763, 18.34311748859386610503749823386, 18.61672812385684634682981073402, 19.77875469588301630052513054249

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