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Properties

Label 1-1259-1259.1036-r0-0-0
Degree $1$
Conductor $1259$
Sign $-0.446 - 0.894i$
Analytic cond. $5.84677$
Root an. cond. $5.84677$
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  + (−0.395 − 0.918i)2-s + (0.999 − 0.0299i)3-s + (−0.686 + 0.727i)4-s + (0.887 + 0.461i)5-s + (−0.423 − 0.906i)6-s + (−0.923 − 0.384i)7-s + (0.939 + 0.342i)8-s + (0.998 − 0.0598i)9-s + (0.0723 − 0.997i)10-s + (−0.414 − 0.910i)11-s + (−0.664 + 0.747i)12-s + (−0.0673 + 0.997i)13-s + (0.0124 + 0.999i)14-s + (0.900 + 0.434i)15-s + (−0.0574 − 0.998i)16-s + (−0.834 − 0.551i)17-s + ⋯
L(s)  = 1  + (−0.395 − 0.918i)2-s + (0.999 − 0.0299i)3-s + (−0.686 + 0.727i)4-s + (0.887 + 0.461i)5-s + (−0.423 − 0.906i)6-s + (−0.923 − 0.384i)7-s + (0.939 + 0.342i)8-s + (0.998 − 0.0598i)9-s + (0.0723 − 0.997i)10-s + (−0.414 − 0.910i)11-s + (−0.664 + 0.747i)12-s + (−0.0673 + 0.997i)13-s + (0.0124 + 0.999i)14-s + (0.900 + 0.434i)15-s + (−0.0574 − 0.998i)16-s + (−0.834 − 0.551i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.446 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.446 - 0.894i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1036, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ -0.446 - 0.894i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8138287407 - 1.316429144i\)
\(L(\frac12)\) \(\approx\) \(0.8138287407 - 1.316429144i\)
\(L(1)\) \(\approx\) \(1.005165951 - 0.5698351983i\)
\(L(1)\) \(\approx\) \(1.005165951 - 0.5698351983i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.395 - 0.918i)T \)
3 \( 1 + (0.999 - 0.0299i)T \)
5 \( 1 + (0.887 + 0.461i)T \)
7 \( 1 + (-0.923 - 0.384i)T \)
11 \( 1 + (-0.414 - 0.910i)T \)
13 \( 1 + (-0.0673 + 0.997i)T \)
17 \( 1 + (-0.834 - 0.551i)T \)
19 \( 1 + (-0.225 - 0.974i)T \)
23 \( 1 + (0.882 - 0.470i)T \)
29 \( 1 + (-0.967 - 0.251i)T \)
31 \( 1 + (-0.870 - 0.492i)T \)
37 \( 1 + (-0.176 - 0.984i)T \)
41 \( 1 + (0.210 - 0.977i)T \)
43 \( 1 + (0.454 - 0.890i)T \)
47 \( 1 + (0.161 - 0.986i)T \)
53 \( 1 + (0.249 + 0.968i)T \)
59 \( 1 + (0.765 - 0.643i)T \)
61 \( 1 + (0.363 - 0.931i)T \)
67 \( 1 + (-0.977 - 0.213i)T \)
71 \( 1 + (0.971 + 0.237i)T \)
73 \( 1 + (0.868 + 0.496i)T \)
79 \( 1 + (-0.511 + 0.859i)T \)
83 \( 1 + (0.445 + 0.895i)T \)
89 \( 1 + (0.996 + 0.0798i)T \)
97 \( 1 + (-0.889 + 0.456i)T \)
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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

−21.22425229347747560969366513972, −20.37941325635211407537435714448, −19.737636842767672749992681961962, −18.96138718914986114962398778136, −18.11443006809465939477874488602, −17.60350868562647189624268902571, −16.56741798111157760946655326039, −15.95392300407823768567161106644, −14.924623878204004422163534594032, −14.84864902494845477482903327560, −13.51377557975265699917786167843, −12.979900800283754144023558602666, −12.64181595966506201893466482939, −10.61002634940994839846411058960, −9.97534414493976737365060004805, −9.383060285311095739952858428431, −8.74917571802453687004280698579, −7.9174059332404874759679104985, −7.074652583755782211366643588982, −6.200871662490118217798639152866, −5.3790533861164845622523458620, −4.49526237344568148076611212987, −3.32967766409904408549358631308, −2.22031252629000800509074796639, −1.32961765921360583114182856482, 0.61372080124379586914922556433, 2.10865710108832741135051020972, 2.49222820900719782396872205060, 3.43524888455651023139950923179, 4.155668384207579120940780649077, 5.40955539663902774953133926196, 6.85243502166990863706214662088, 7.19425686663773398554364105003, 8.581564063595802059939522096729, 9.28124351245982080178750057309, 9.52259432950322787463599932379, 10.76074275078334899464221223094, 11.01476811352002901404301775985, 12.485582541713682940031848692311, 13.278885209473219224411739581598, 13.60799939691656652712481470629, 14.26291274672763037713223757178, 15.41438631860176146827087809684, 16.39510850139798378540255889726, 17.07597994394230295962997371752, 18.08930643107825954500457097789, 18.93500502375230852011217980521, 19.056437604244610412026694542370, 20.09984864518764614256545082224, 20.694299759312696566852951046205

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