Properties

Label 2-1028-1028.15-c0-0-0
Degree $2$
Conductor $1028$
Sign $0.0543 - 0.998i$
Analytic cond. $0.513038$
Root an. cond. $0.716267$
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  + i·2-s − 4-s + (1.75 + 0.172i)5-s i·8-s + (0.195 + 0.980i)9-s + (−0.172 + 1.75i)10-s + (−1.63 − 1.08i)13-s + 16-s + (1.17 + 1.17i)17-s + (−0.980 + 0.195i)18-s + (−1.75 − 0.172i)20-s + (2.07 + 0.411i)25-s + (1.08 − 1.63i)26-s + (0.275 − 1.38i)29-s + i·32-s + ⋯
L(s)  = 1  + i·2-s − 4-s + (1.75 + 0.172i)5-s i·8-s + (0.195 + 0.980i)9-s + (−0.172 + 1.75i)10-s + (−1.63 − 1.08i)13-s + 16-s + (1.17 + 1.17i)17-s + (−0.980 + 0.195i)18-s + (−1.75 − 0.172i)20-s + (2.07 + 0.411i)25-s + (1.08 − 1.63i)26-s + (0.275 − 1.38i)29-s + i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0543 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1028\)    =    \(2^{2} \cdot 257\)
Sign: $0.0543 - 0.998i$
Analytic conductor: \(0.513038\)
Root analytic conductor: \(0.716267\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1028} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1028,\ (\ :0),\ 0.0543 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.195983737\)
\(L(\frac12)\) \(\approx\) \(1.195983737\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
257 \( 1 + T \)
good3 \( 1 + (-0.195 - 0.980i)T^{2} \)
5 \( 1 + (-1.75 - 0.172i)T + (0.980 + 0.195i)T^{2} \)
7 \( 1 + (0.831 + 0.555i)T^{2} \)
11 \( 1 + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \)
17 \( 1 + (-1.17 - 1.17i)T + iT^{2} \)
19 \( 1 + (0.555 - 0.831i)T^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.275 + 1.38i)T + (-0.923 - 0.382i)T^{2} \)
31 \( 1 + (-0.923 + 0.382i)T^{2} \)
37 \( 1 + (1.68 - 0.512i)T + (0.831 - 0.555i)T^{2} \)
41 \( 1 + (0.172 + 0.0924i)T + (0.555 + 0.831i)T^{2} \)
43 \( 1 + (0.195 - 0.980i)T^{2} \)
47 \( 1 + (0.555 - 0.831i)T^{2} \)
53 \( 1 + (-0.0924 - 0.938i)T + (-0.980 + 0.195i)T^{2} \)
59 \( 1 + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (0.636 + 0.425i)T + (0.382 + 0.923i)T^{2} \)
67 \( 1 + (-0.707 + 0.707i)T^{2} \)
71 \( 1 + (0.555 + 0.831i)T^{2} \)
73 \( 1 + (0.149 - 0.360i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + (-0.382 + 0.923i)T^{2} \)
83 \( 1 + (0.195 - 0.980i)T^{2} \)
89 \( 1 + (-0.216 - 0.324i)T + (-0.382 + 0.923i)T^{2} \)
97 \( 1 + (-0.124 + 1.26i)T + (-0.980 - 0.195i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.979310699679220345022125565816, −9.801503453350175679852620314410, −8.512704714484017527292469044138, −7.78900125502211549583741121805, −6.96392982068639468579323552854, −5.90348583613563610142928354928, −5.44546474334347280209394132388, −4.67852473723154581115201394965, −3.02355117026895088053723520843, −1.78372075867876704961122071100, 1.37828472835755954226645719106, 2.37532307034652720143064142655, 3.36486195655928157787210274108, 4.86131278208250810592220373818, 5.29356996044291923697629528919, 6.47861535646834333911486143956, 7.35567292295286412448060969693, 8.930672575222785415559444263279, 9.320740764474637625981604597843, 9.903073542732809715588608717237

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