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Properties

Label 1-1259-1259.1009-r0-0-0
Degree $1$
Conductor $1259$
Sign $-0.982 - 0.186i$
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  + (−0.414 − 0.910i)2-s + (−0.997 − 0.0648i)3-s + (−0.656 + 0.753i)4-s + (0.507 − 0.861i)5-s + (0.354 + 0.935i)6-s + (0.191 − 0.981i)7-s + (0.958 + 0.285i)8-s + (0.991 + 0.129i)9-s + (−0.994 − 0.104i)10-s + (0.990 + 0.139i)11-s + (0.704 − 0.709i)12-s + (0.802 + 0.596i)13-s + (−0.972 + 0.232i)14-s + (−0.562 + 0.827i)15-s + (−0.136 − 0.990i)16-s + (−0.953 − 0.299i)17-s + ⋯
L(s)  = 1  + (−0.414 − 0.910i)2-s + (−0.997 − 0.0648i)3-s + (−0.656 + 0.753i)4-s + (0.507 − 0.861i)5-s + (0.354 + 0.935i)6-s + (0.191 − 0.981i)7-s + (0.958 + 0.285i)8-s + (0.991 + 0.129i)9-s + (−0.994 − 0.104i)10-s + (0.990 + 0.139i)11-s + (0.704 − 0.709i)12-s + (0.802 + 0.596i)13-s + (−0.972 + 0.232i)14-s + (−0.562 + 0.827i)15-s + (−0.136 − 0.990i)16-s + (−0.953 − 0.299i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $-0.982 - 0.186i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ -0.982 - 0.186i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07875412259 - 0.8384421608i\)
\(L(\frac12)\) \(\approx\) \(0.07875412259 - 0.8384421608i\)
\(L(1)\) \(\approx\) \(0.5274270104 - 0.4782925955i\)
\(L(1)\) \(\approx\) \(0.5274270104 - 0.4782925955i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.414 - 0.910i)T \)
3 \( 1 + (-0.997 - 0.0648i)T \)
5 \( 1 + (0.507 - 0.861i)T \)
7 \( 1 + (0.191 - 0.981i)T \)
11 \( 1 + (0.990 + 0.139i)T \)
13 \( 1 + (0.802 + 0.596i)T \)
17 \( 1 + (-0.953 - 0.299i)T \)
19 \( 1 + (-0.728 - 0.684i)T \)
23 \( 1 + (-0.511 + 0.859i)T \)
29 \( 1 + (0.524 + 0.851i)T \)
31 \( 1 + (-0.0673 - 0.997i)T \)
37 \( 1 + (0.122 - 0.992i)T \)
41 \( 1 + (0.660 + 0.750i)T \)
43 \( 1 + (-0.235 - 0.971i)T \)
47 \( 1 + (0.995 - 0.0897i)T \)
53 \( 1 + (-0.959 - 0.280i)T \)
59 \( 1 + (0.836 - 0.547i)T \)
61 \( 1 + (-0.855 + 0.517i)T \)
67 \( 1 + (0.549 - 0.835i)T \)
71 \( 1 + (0.868 - 0.496i)T \)
73 \( 1 + (0.566 + 0.824i)T \)
79 \( 1 + (0.784 + 0.620i)T \)
83 \( 1 + (0.739 + 0.673i)T \)
89 \( 1 + (-0.641 - 0.766i)T \)
97 \( 1 + (-0.875 - 0.483i)T \)
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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

−21.857972047376346284673542428262, −20.86498675539358527942423281252, −19.465370883224973445096505815579, −18.771491387300326144031009978524, −18.15546372379512991208499533308, −17.61417909087821327182171466001, −17.00704214643524395181049294911, −16.02093866053291584551709303114, −15.40863355771878988357803951657, −14.72924613217744670900690499820, −13.92679310588889034078432039975, −12.95577172955738599293734077690, −12.04246141851330919527108312781, −11.00711933509635735073747185760, −10.52406164610655968518942459177, −9.62945369000615936489801597459, −8.77065541884225653337215402085, −7.957495903966052749981070024933, −6.645395700096422370824035976377, −6.29669870793231837989049691095, −5.80806152758945561044101430566, −4.73675415693208111503997172128, −3.796865636910055447557599733133, −2.21340000868420651553878590150, −1.20524900453630529971435822995, 0.51922352331120780664292273389, 1.378455727720885350687878528975, 2.07369941013757251364859803753, 3.94739467483352258374262986923, 4.24744373482538498581157334830, 5.15864838320559641603946879954, 6.36072655670248776267857029317, 7.09370741411181166582794391694, 8.22525983868067236395096984788, 9.21373518566947830694463750021, 9.66032577102344583205130382759, 10.826515768227762580075319835892, 11.16651286283399905030288060090, 12.02928003196239671289266943619, 12.82196159969654961112333658510, 13.50792374841588278072360539527, 14.068592026992845435328681163579, 15.69298183081975476283579596076, 16.568109060775139889790160292090, 17.01855725866884260113441690927, 17.66554268625579786884437446527, 18.16261528789337830431042363794, 19.354456824917216288739394618542, 19.95313769704491179135258641736, 20.679030049979685827767534625804

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