↑

Properties

Label 1-1259-1259.1048-r0-0-0
Degree $1$
Conductor $1259$
Sign $0.0757 - 0.997i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.445 − 0.895i)2-s + (−0.850 − 0.526i)3-s + (−0.602 − 0.798i)4-s + (−0.850 − 0.526i)5-s + (−0.850 + 0.526i)6-s + (−0.850 − 0.526i)7-s + (−0.982 + 0.183i)8-s + (0.445 + 0.895i)9-s + (−0.850 + 0.526i)10-s + (0.739 − 0.673i)11-s + (0.0922 + 0.995i)12-s + (0.445 + 0.895i)13-s + (−0.850 + 0.526i)14-s + (0.445 + 0.895i)15-s + (−0.273 + 0.961i)16-s + (−0.982 − 0.183i)17-s + ⋯
L(s)  = 1  + (0.445 − 0.895i)2-s + (−0.850 − 0.526i)3-s + (−0.602 − 0.798i)4-s + (−0.850 − 0.526i)5-s + (−0.850 + 0.526i)6-s + (−0.850 − 0.526i)7-s + (−0.982 + 0.183i)8-s + (0.445 + 0.895i)9-s + (−0.850 + 0.526i)10-s + (0.739 − 0.673i)11-s + (0.0922 + 0.995i)12-s + (0.445 + 0.895i)13-s + (−0.850 + 0.526i)14-s + (0.445 + 0.895i)15-s + (−0.273 + 0.961i)16-s + (−0.982 − 0.183i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0757 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0757 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1259\)
Sign: $0.0757 - 0.997i$
Analytic conductor: \(5.84677\)
Root analytic conductor: \(5.84677\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1259} (1048, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1259,\ (0:\ ),\ 0.0757 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5872523476 - 0.5443590948i\)
\(L(\frac12)\) \(\approx\) \(0.5872523476 - 0.5443590948i\)
\(L(1)\) \(\approx\) \(0.5625218105 - 0.4894143835i\)
\(L(1)\) \(\approx\) \(0.5625218105 - 0.4894143835i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad1259 \( 1 \)
good2 \( 1 + (0.445 - 0.895i)T \)
3 \( 1 + (-0.850 - 0.526i)T \)
5 \( 1 + (-0.850 - 0.526i)T \)
7 \( 1 + (-0.850 - 0.526i)T \)
11 \( 1 + (0.739 - 0.673i)T \)
13 \( 1 + (0.445 + 0.895i)T \)
17 \( 1 + (-0.982 - 0.183i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.932 - 0.361i)T \)
29 \( 1 + T \)
31 \( 1 + (0.0922 + 0.995i)T \)
37 \( 1 + (-0.602 - 0.798i)T \)
41 \( 1 + T \)
43 \( 1 + (0.0922 + 0.995i)T \)
47 \( 1 + (-0.602 - 0.798i)T \)
53 \( 1 + (0.739 + 0.673i)T \)
59 \( 1 + (0.932 + 0.361i)T \)
61 \( 1 + (-0.982 - 0.183i)T \)
67 \( 1 + (0.739 + 0.673i)T \)
71 \( 1 + (-0.273 + 0.961i)T \)
73 \( 1 + (-0.982 + 0.183i)T \)
79 \( 1 + (-0.273 + 0.961i)T \)
83 \( 1 + (0.0922 + 0.995i)T \)
89 \( 1 + (0.0922 - 0.995i)T \)
97 \( 1 + (-0.602 - 0.798i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.70345321838104008240726189970, −20.642682423374847543459555874166, −19.64822467440797556436324121672, −18.82161482997217219749938068170, −17.77766404110970025065963037996, −17.46563131451479813998050452435, −16.421457665513454918516837016446, −15.67941493004522927398703397167, −15.29268046312360909503252911652, −14.82505737707266172961341578278, −13.42158637102176489665256205642, −12.72169060225167195523831443225, −11.98494752783441285284129644440, −11.28381433194227413230105157124, −10.32129252264647292892605205188, −9.28951675660298597778545866592, −8.65639918959484021025012793799, −7.43049923735200231469917682546, −6.61107471246120577142661666014, −6.27814185950569938064517396321, −5.12789023624185100495658285279, −4.36188628093005455178430414453, −3.59044332557562244857440831547, −2.75421362953959452033609161679, −0.49117369367382314557044781042, 0.76874306440429197434768172769, 1.48000828067192937957575417311, 2.883597886705010944262156203080, 4.005210132576462807394406308960, 4.38590601495760140863416397916, 5.52732653806500058993429365644, 6.479599849934660275249240740795, 6.99915379767873252780958847420, 8.46086209364897902915743777629, 9.07494330764943770263797629917, 10.22500832232028030936732918518, 11.04259762263364263113647848186, 11.53571318626373746711957328164, 12.36190411172368082502274653177, 12.86968247336536035246898176090, 13.67593308347368477112011399075, 14.386970355412645194884621525126, 15.73213574724668935078322575955, 16.32188477996360788006810797901, 16.99399488832002922102894605657, 18.01228155200833266853121217675, 18.934102738974469865415233116621, 19.42291633512077021152138898982, 19.84028785373960637343345082567, 20.979365689920829160360402819420

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