↑

Properties

Label 1-2557-2557.42-r1-0-0
Degree $1$
Conductor $2557$
Sign $0.953 - 0.300i$
Analytic cond. $274.787$
Root an. cond. $274.787$
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  + (−0.122 − 0.992i)2-s + (−0.936 + 0.351i)3-s + (−0.969 + 0.243i)4-s + (0.679 − 0.733i)5-s + (0.463 + 0.886i)6-s + (−0.941 − 0.337i)7-s + (0.360 + 0.932i)8-s + (0.753 − 0.657i)9-s + (−0.811 − 0.584i)10-s + (−0.236 − 0.971i)11-s + (0.822 − 0.568i)12-s + (0.399 − 0.916i)13-s + (−0.219 + 0.975i)14-s + (−0.378 + 0.925i)15-s + (0.881 − 0.472i)16-s + (0.619 + 0.784i)17-s + ⋯
L(s)  = 1  + (−0.122 − 0.992i)2-s + (−0.936 + 0.351i)3-s + (−0.969 + 0.243i)4-s + (0.679 − 0.733i)5-s + (0.463 + 0.886i)6-s + (−0.941 − 0.337i)7-s + (0.360 + 0.932i)8-s + (0.753 − 0.657i)9-s + (−0.811 − 0.584i)10-s + (−0.236 − 0.971i)11-s + (0.822 − 0.568i)12-s + (0.399 − 0.916i)13-s + (−0.219 + 0.975i)14-s + (−0.378 + 0.925i)15-s + (0.881 − 0.472i)16-s + (0.619 + 0.784i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2557 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.953 - 0.300i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2557\)
Sign: $0.953 - 0.300i$
Analytic conductor: \(274.787\)
Root analytic conductor: \(274.787\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2557} (42, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2557,\ (1:\ ),\ 0.953 - 0.300i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8973197233 - 0.1381559050i\)
\(L(\frac12)\) \(\approx\) \(0.8973197233 - 0.1381559050i\)
\(L(1)\) \(\approx\) \(0.6016998847 - 0.3797842851i\)
\(L(1)\) \(\approx\) \(0.6016998847 - 0.3797842851i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2557 \( 1 \)
good2 \( 1 + (-0.122 - 0.992i)T \)
3 \( 1 + (-0.936 + 0.351i)T \)
5 \( 1 + (0.679 - 0.733i)T \)
7 \( 1 + (-0.941 - 0.337i)T \)
11 \( 1 + (-0.236 - 0.971i)T \)
13 \( 1 + (0.399 - 0.916i)T \)
17 \( 1 + (0.619 + 0.784i)T \)
19 \( 1 + (0.999 - 0.0393i)T \)
23 \( 1 + (-0.994 - 0.107i)T \)
29 \( 1 + (-0.976 + 0.214i)T \)
31 \( 1 + (0.988 - 0.149i)T \)
37 \( 1 + (0.999 + 0.00491i)T \)
41 \( 1 + (0.789 + 0.614i)T \)
43 \( 1 + (-0.0344 - 0.999i)T \)
47 \( 1 + (-0.916 + 0.399i)T \)
53 \( 1 + (0.952 - 0.304i)T \)
59 \( 1 + (0.859 + 0.510i)T \)
61 \( 1 + (0.633 + 0.773i)T \)
67 \( 1 + (-0.997 + 0.0761i)T \)
71 \( 1 + (0.570 + 0.821i)T \)
73 \( 1 + (0.348 - 0.937i)T \)
79 \( 1 + (-0.387 + 0.921i)T \)
83 \( 1 + (0.942 - 0.335i)T \)
89 \( 1 + (0.489 + 0.872i)T \)
97 \( 1 + (-0.720 + 0.693i)T \)
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

−18.84513162182480988312805253875, −18.275895583605084661211239981622, −17.96442688661815491865440109477, −17.13584719722110479999047727354, −16.30470981480996417386035396803, −16.01324174344346371897653948805, −15.11991444098091235837634514703, −14.254047903051906338727990735102, −13.579687670347792829730204212784, −13.014219339606594919040285284799, −12.13625340814401335731073496413, −11.4213411178975504446195279248, −10.288234153925943741558979579214, −9.66985970673819794366959975011, −9.42007512219808599366590269293, −7.94403310585830841586692671330, −7.21750800035124319560022676456, −6.7199287183916328995975663024, −6.010500095264799008556603753040, −5.516109809887334594969354923110, −4.61574741624991740586192790131, −3.627180424265274250528338492045, −2.42310839601066973524300243863, −1.421718758397023714344940514673, −0.261570243639875121043956816085, 0.845994014989519811365823866860, 1.03295271881874399974128275842, 2.475123239815032508340842306994, 3.49460378553040571227828801858, 3.99577128870287041471126555258, 5.11468393906665579495216893759, 5.76258560479626542592066390068, 6.17660273652898329589375334119, 7.64330316809458813234173674626, 8.4569705397915923589327209822, 9.35830716276127569111850389616, 10.017777065601111320925093418250, 10.350582384776375774090950214593, 11.221792508558666377023445304877, 12.00160738782136050696887365940, 12.69882212911822648755491554293, 13.246236939789084749253387814831, 13.74996159220049725858356411266, 14.89994344958824962185533159838, 16.14774014893937172938940105714, 16.37333396505384827887725345575, 17.12477079162311697685213409489, 17.87609081486249135187404166171, 18.36271603764378906626434011687, 19.25506943750964322856016558354

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