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Properties

Label 1-1259-1259.1090-r1-0-0
Degree $1$
Conductor $1259$
Sign $0.992 - 0.120i$
Analytic cond. $135.298$
Root an. cond. $135.298$
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  + (−0.877 − 0.478i)2-s + (0.718 + 0.695i)3-s + (0.541 + 0.840i)4-s + (0.966 + 0.256i)5-s + (−0.297 − 0.954i)6-s + (−0.941 − 0.337i)7-s + (−0.0723 − 0.997i)8-s + (0.0324 + 0.999i)9-s + (−0.725 − 0.688i)10-s + (0.999 − 0.0349i)11-s + (−0.196 + 0.980i)12-s + (−0.987 + 0.159i)13-s + (0.664 + 0.747i)14-s + (0.515 + 0.856i)15-s + (−0.414 + 0.910i)16-s + (0.758 + 0.651i)17-s + ⋯
L(s)  = 1  + (−0.877 − 0.478i)2-s + (0.718 + 0.695i)3-s + (0.541 + 0.840i)4-s + (0.966 + 0.256i)5-s + (−0.297 − 0.954i)6-s + (−0.941 − 0.337i)7-s + (−0.0723 − 0.997i)8-s + (0.0324 + 0.999i)9-s + (−0.725 − 0.688i)10-s + (0.999 − 0.0349i)11-s + (−0.196 + 0.980i)12-s + (−0.987 + 0.159i)13-s + (0.664 + 0.747i)14-s + (0.515 + 0.856i)15-s + (−0.414 + 0.910i)16-s + (0.758 + 0.651i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.992 - 0.120i$
Analytic conductor: \(135.298\)
Root analytic conductor: \(135.298\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1090, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (1:\ ),\ 0.992 - 0.120i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.154763782 - 0.1304259430i\)
\(L(\frac12)\) \(\approx\) \(2.154763782 - 0.1304259430i\)
\(L(1)\) \(\approx\) \(1.077905462 + 0.03607568248i\)
\(L(1)\) \(\approx\) \(1.077905462 + 0.03607568248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (-0.877 - 0.478i)T \)
3 \( 1 + (0.718 + 0.695i)T \)
5 \( 1 + (0.966 + 0.256i)T \)
7 \( 1 + (-0.941 - 0.337i)T \)
11 \( 1 + (0.999 - 0.0349i)T \)
13 \( 1 + (-0.987 + 0.159i)T \)
17 \( 1 + (0.758 + 0.651i)T \)
19 \( 1 + (0.562 - 0.827i)T \)
23 \( 1 + (-0.502 - 0.864i)T \)
29 \( 1 + (0.967 - 0.251i)T \)
31 \( 1 + (-0.917 - 0.398i)T \)
37 \( 1 + (0.354 - 0.935i)T \)
41 \( 1 + (-0.210 - 0.977i)T \)
43 \( 1 + (0.436 - 0.899i)T \)
47 \( 1 + (-0.0224 + 0.999i)T \)
53 \( 1 + (-0.755 - 0.655i)T \)
59 \( 1 + (0.989 + 0.144i)T \)
61 \( 1 + (0.796 - 0.604i)T \)
67 \( 1 + (-0.244 + 0.969i)T \)
71 \( 1 + (0.991 + 0.129i)T \)
73 \( 1 + (0.239 + 0.970i)T \)
79 \( 1 + (-0.166 - 0.986i)T \)
83 \( 1 + (-0.982 + 0.183i)T \)
89 \( 1 + (0.537 - 0.843i)T \)
97 \( 1 + (0.790 + 0.612i)T \)
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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

−20.55175674111972618641317363525, −19.78361682236395535344874294941, −19.45855219204312867517825293503, −18.38952845558083785828713193685, −18.06738564109870272479532169406, −17.04531174988623562639105387699, −16.53506326807338060988009296942, −15.59528816655098515804329307172, −14.49106613094399599339129051240, −14.24881490198300067812193090027, −13.27405172771836503630005789096, −12.28308242402691887005955530085, −11.76192935561757189521236282417, −10.10652153362520382467672399054, −9.63948451329701548675277703053, −9.22619509305463873734879758158, −8.25636107639556208032088957880, −7.38503476430250069318693562103, −6.615818450598517206949231158614, −5.99783336578928179770937515567, −5.10002376541631630697766625871, −3.38337564127684205345046139159, −2.552528638223738675915937926693, −1.59023753821915699747169970690, −0.847708239261013283039601409917, 0.63084936619937753744415857744, 1.94064650058107950209659634426, 2.64924940893155884927618220936, 3.488392221054266822922762738877, 4.30318480681636938879722900331, 5.69212751889529328643459321667, 6.75380671173226409117448742879, 7.38320512753232444831063932201, 8.570220368428623331188836126616, 9.30063446850140593100924035159, 9.81327118505151657328392776782, 10.29783026948662941225991001885, 11.177836121605810117130889523306, 12.35484419948672420083772349806, 13.00930913256536099941292657445, 14.07840780815168342697032068900, 14.54286999916346326171785692990, 15.70903721385929909420601665142, 16.441296375658918962855787101589, 17.07398328465369206580363080520, 17.66846965676253134950368242217, 18.928837199388291769145989840106, 19.30432770666319701990030952722, 20.074978027969953223339977301529, 20.64462138355350357387388264950

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