↑

Properties

Label 1-1027-1027.119-r1-0-0
Degree $1$
Conductor $1027$
Sign $0.620 + 0.784i$
Analytic cond. $110.366$
Root an. cond. $110.366$
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  + (0.721 − 0.692i)2-s + (−0.200 + 0.979i)3-s + (0.0402 − 0.999i)4-s + (−0.903 − 0.428i)5-s + (0.534 + 0.845i)6-s + (0.316 + 0.948i)7-s + (−0.663 − 0.748i)8-s + (−0.919 − 0.391i)9-s + (−0.948 + 0.316i)10-s + (0.822 + 0.568i)11-s + (0.970 + 0.239i)12-s + (0.885 + 0.464i)14-s + (0.600 − 0.799i)15-s + (−0.996 − 0.0804i)16-s + (0.845 + 0.534i)17-s + (−0.935 + 0.354i)18-s + ⋯
L(s)  = 1  + (0.721 − 0.692i)2-s + (−0.200 + 0.979i)3-s + (0.0402 − 0.999i)4-s + (−0.903 − 0.428i)5-s + (0.534 + 0.845i)6-s + (0.316 + 0.948i)7-s + (−0.663 − 0.748i)8-s + (−0.919 − 0.391i)9-s + (−0.948 + 0.316i)10-s + (0.822 + 0.568i)11-s + (0.970 + 0.239i)12-s + (0.885 + 0.464i)14-s + (0.600 − 0.799i)15-s + (−0.996 − 0.0804i)16-s + (0.845 + 0.534i)17-s + (−0.935 + 0.354i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.620 + 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.620 + 0.784i$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 0.620 + 0.784i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.739635074 + 0.8420839412i\)
\(L(\frac12)\) \(\approx\) \(1.739635074 + 0.8420839412i\)
\(L(1)\) \(\approx\) \(1.226924686 - 0.06933318480i\)
\(L(1)\) \(\approx\) \(1.226924686 - 0.06933318480i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (0.721 - 0.692i)T \)
3 \( 1 + (-0.200 + 0.979i)T \)
5 \( 1 + (-0.903 - 0.428i)T \)
7 \( 1 + (0.316 + 0.948i)T \)
11 \( 1 + (0.822 + 0.568i)T \)
17 \( 1 + (0.845 + 0.534i)T \)
19 \( 1 + (0.160 - 0.987i)T \)
23 \( 1 - T \)
29 \( 1 + (0.120 - 0.992i)T \)
31 \( 1 + (0.721 - 0.692i)T \)
37 \( 1 + (-0.774 - 0.632i)T \)
41 \( 1 + (0.0804 + 0.996i)T \)
43 \( 1 + (-0.568 - 0.822i)T \)
47 \( 1 + (0.160 + 0.987i)T \)
53 \( 1 + (0.948 + 0.316i)T \)
59 \( 1 + (-0.464 + 0.885i)T \)
61 \( 1 + (-0.632 + 0.774i)T \)
67 \( 1 + (-0.721 - 0.692i)T \)
71 \( 1 + (0.316 - 0.948i)T \)
73 \( 1 + (0.999 - 0.0402i)T \)
83 \( 1 + (-0.534 + 0.845i)T \)
89 \( 1 + (0.316 + 0.948i)T \)
97 \( 1 + (0.774 + 0.632i)T \)
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

−21.48368875595253083120658643404, −20.349604643366308055205310497839, −19.825374029584449845487635239157, −18.81920498936353329110464312612, −18.135442565814467608176271848959, −17.17824906508801902130001858702, −16.55873453569240060268846346279, −15.912442210519881065989917123181, −14.599825125018547212473860353111, −14.18081260530520421950198423459, −13.66178735021824556658821501741, −12.43321182621956736112186073731, −11.93696310151164360650362871405, −11.30202495639796044816240976090, −10.254237158680825719609575682461, −8.55891607143837031781789621291, −8.04396354698677359723377743265, −7.22339968661994698134021271508, −6.73872332069224066309612375785, −5.805489079221529783918643874952, −4.77130022146129385092705905101, −3.67163413094691096018553455906, −3.16421265490664588983097029910, −1.6018127663065975433766894031, −0.398403830296384746738670755131, 0.889771553563443231257980546065, 2.217688523930168594268650306012, 3.28042376340709857719522113094, 4.14257672875608367091592990764, 4.6857629916672441811672342940, 5.57975754887449898329666893947, 6.37706240950011615512854300313, 7.85183408097193992342462392755, 8.90882690186328333355116234448, 9.4943130081996246073429575671, 10.42293875430013750308888625605, 11.38342897174019099314680802150, 11.99067503240281108497966388818, 12.292493843566839910520776735083, 13.62818150233940698268797746206, 14.61780908576542349927224423678, 15.22968221091032817378383422190, 15.624716025825910460918364852456, 16.62434796815190683410192103415, 17.574687711511478483029475826039, 18.60450010132418153536452864821, 19.64373254856428880929265483561, 19.90246760040280982935084343584, 21.04193478982877437386510863642, 21.30088399913893022344632268418

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