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Properties

Label 1-919-919.535-r0-0-0
Degree $1$
Conductor $919$
Sign $0.0746 + 0.997i$
Analytic cond. $4.26781$
Root an. cond. $4.26781$
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  + (−0.982 + 0.183i)2-s + (−0.850 + 0.526i)3-s + (0.932 − 0.361i)4-s + (−0.850 + 0.526i)5-s + (0.739 − 0.673i)6-s + (0.739 − 0.673i)7-s + (−0.850 + 0.526i)8-s + (0.445 − 0.895i)9-s + (0.739 − 0.673i)10-s + (0.445 + 0.895i)11-s + (−0.602 + 0.798i)12-s + (0.932 + 0.361i)13-s + (−0.602 + 0.798i)14-s + (0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.273 + 0.961i)17-s + ⋯
L(s)  = 1  + (−0.982 + 0.183i)2-s + (−0.850 + 0.526i)3-s + (0.932 − 0.361i)4-s + (−0.850 + 0.526i)5-s + (0.739 − 0.673i)6-s + (0.739 − 0.673i)7-s + (−0.850 + 0.526i)8-s + (0.445 − 0.895i)9-s + (0.739 − 0.673i)10-s + (0.445 + 0.895i)11-s + (−0.602 + 0.798i)12-s + (0.932 + 0.361i)13-s + (−0.602 + 0.798i)14-s + (0.445 − 0.895i)15-s + (0.739 − 0.673i)16-s + (−0.273 + 0.961i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0746 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0746 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(919\)
Sign: $0.0746 + 0.997i$
Analytic conductor: \(4.26781\)
Root analytic conductor: \(4.26781\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{919} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 919,\ (0:\ ),\ 0.0746 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4628813370 + 0.4295108004i\)
\(L(\frac12)\) \(\approx\) \(0.4628813370 + 0.4295108004i\)
\(L(1)\) \(\approx\) \(0.5196385973 + 0.1928362691i\)
\(L(1)\) \(\approx\) \(0.5196385973 + 0.1928362691i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad919 \( 1 \)
good2 \( 1 + (-0.982 + 0.183i)T \)
3 \( 1 + (-0.850 + 0.526i)T \)
5 \( 1 + (-0.850 + 0.526i)T \)
7 \( 1 + (0.739 - 0.673i)T \)
11 \( 1 + (0.445 + 0.895i)T \)
13 \( 1 + (0.932 + 0.361i)T \)
17 \( 1 + (-0.273 + 0.961i)T \)
19 \( 1 + (-0.850 - 0.526i)T \)
23 \( 1 + (0.445 + 0.895i)T \)
29 \( 1 + (-0.982 + 0.183i)T \)
31 \( 1 + (0.739 - 0.673i)T \)
37 \( 1 + (0.932 + 0.361i)T \)
41 \( 1 + (-0.602 - 0.798i)T \)
43 \( 1 + (-0.850 - 0.526i)T \)
47 \( 1 + (0.932 - 0.361i)T \)
53 \( 1 + T \)
59 \( 1 + (0.932 - 0.361i)T \)
61 \( 1 + (0.739 + 0.673i)T \)
67 \( 1 + (0.0922 + 0.995i)T \)
71 \( 1 + (0.739 - 0.673i)T \)
73 \( 1 + (0.0922 + 0.995i)T \)
79 \( 1 + (0.739 + 0.673i)T \)
83 \( 1 + (-0.273 - 0.961i)T \)
89 \( 1 + (-0.850 - 0.526i)T \)
97 \( 1 + (0.445 - 0.895i)T \)
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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

−21.49475425762497539373463584948, −20.83175874970179677388801463018, −19.967405824900653412586585750071, −19.00808470424594504622072658797, −18.56881516318697538790302996395, −17.92326268848484381139198883504, −16.87656649958599455390650292340, −16.436540749208381993995479235500, −15.67509276261733335974558057042, −14.79458366062489300506389721443, −13.36918093850682321194752138342, −12.48903114242913473866763070025, −11.75920248927513055884027978392, −11.2290387786555028961234966427, −10.67396467222907314465118317500, −9.24262299257765322163119783253, −8.34576412435636139999784349856, −8.054477546144765896902771220834, −6.86703490161498173033767206254, −6.07342616586971615913323998870, −5.11016248459556843760877849392, −3.93604898406488620742162748594, −2.62566934892308261179142961216, −1.4184426542238783022819037892, −0.61754395107977854637985356582, 0.92197760217686734209111079007, 2.01417699594997455313547000348, 3.733781278247438193816174014948, 4.27647234462602103036492862376, 5.54423462723267515112341728647, 6.69674912476747878669049889320, 7.066129939869570840678320905999, 8.14844066321852222604202937677, 8.97935723912407538228043239496, 10.10226007367320663871365060395, 10.70371522050399729444614083215, 11.41059187839285636035031656312, 11.80466486762354389190210474757, 13.11211058700488555092323875424, 14.61895987772417470482488454103, 15.18273026244329423862092736034, 15.6772387113340657990514221623, 16.91831851204666932685813516248, 17.11734212237336136551228998185, 18.03029726277529506362969299346, 18.73683461921656624223249578539, 19.66015565908672861945183105105, 20.41041679262641551785707100560, 21.1242665390607115561131627661, 22.0627564078762655019137396356

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