↑

Properties

Label 1-2e7-128.77-r0-0-0
Degree $1$
Conductor $128$
Sign $0.671 - 0.740i$
Analytic cond. $0.594429$
Root an. cond. $0.594429$
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  + (0.831 − 0.555i)3-s + (0.980 − 0.195i)5-s + (−0.382 − 0.923i)7-s + (0.382 − 0.923i)9-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.195 + 0.980i)19-s + (−0.831 − 0.555i)21-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.195 − 0.980i)27-s + (0.555 + 0.831i)29-s − i·31-s + ⋯
L(s)  = 1  + (0.831 − 0.555i)3-s + (0.980 − 0.195i)5-s + (−0.382 − 0.923i)7-s + (0.382 − 0.923i)9-s + (−0.555 + 0.831i)11-s + (−0.980 − 0.195i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.195 + 0.980i)19-s + (−0.831 − 0.555i)21-s + (−0.923 − 0.382i)23-s + (0.923 − 0.382i)25-s + (−0.195 − 0.980i)27-s + (0.555 + 0.831i)29-s − i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(0.594429\)
Root analytic conductor: \(0.594429\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (0:\ ),\ 0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.337254980 - 0.5927643650i\)
\(L(\frac12)\) \(\approx\) \(1.337254980 - 0.5927643650i\)
\(L(1)\) \(\approx\) \(1.337270028 - 0.3641475505i\)
\(L(1)\) \(\approx\) \(1.337270028 - 0.3641475505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 + (0.831 - 0.555i)T \)
5 \( 1 + (0.980 - 0.195i)T \)
7 \( 1 + (-0.382 - 0.923i)T \)
11 \( 1 + (-0.555 + 0.831i)T \)
13 \( 1 + (-0.980 - 0.195i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.195 + 0.980i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (0.555 + 0.831i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.195 + 0.980i)T \)
41 \( 1 + (0.923 + 0.382i)T \)
43 \( 1 + (0.831 + 0.555i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.555 - 0.831i)T \)
59 \( 1 + (-0.980 + 0.195i)T \)
61 \( 1 + (-0.831 + 0.555i)T \)
67 \( 1 + (-0.831 + 0.555i)T \)
71 \( 1 + (0.382 + 0.923i)T \)
73 \( 1 + (-0.382 + 0.923i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.195 - 0.980i)T \)
89 \( 1 + (-0.923 + 0.382i)T \)
97 \( 1 - iT \)
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

−28.96225101338881678579775841229, −27.89517455282916922506856766314, −26.69867024736633015450422627743, −25.938619941918543280097486607259, −25.078969751924154479456700146374, −24.299379488012619255105979563600, −22.522441668587860231253607721709, −21.53745449682527477267646761203, −21.24899490948012372458099717643, −19.72892268129491318574046117725, −18.88161472046944638610635520264, −17.783497630848748912365529306539, −16.37585468812695530740671051473, −15.51076559951530180263356003185, −14.317540711309922119106703988278, −13.60693104773427119607211314259, −12.33742387219431992913353041647, −10.73022418183942361130503095154, −9.62968933538310081347202237429, −8.98614263131245775439077926522, −7.5872915425903234462175847628, −5.9660037516336673067402898175, −4.90357451034973786252082361387, −3.02917539976888220611533785495, −2.30002001360756392232798231650, 1.53063937773295947894573216440, 2.767875881492242080784076817885, 4.32291196713201797191445166278, 5.9997579082149972501996522621, 7.2229505242298639268759190904, 8.19294182790402594986478487516, 9.8065376202195123236495068420, 10.15799833133184205794603673098, 12.40371525207330585373432300363, 13.00367563743483014176965507101, 14.12526449327614332942287895733, 14.8448768938024869073337268346, 16.500151173904715578376128933655, 17.49379828128221177262108104011, 18.44617112072760731695763054159, 19.68625974147260432575357421783, 20.4552217079564136689615060489, 21.33862882842768364198532012618, 22.72178969668452232448561368593, 23.82822860808679065896589633372, 24.76131954155858547974721155212, 25.82765462736102299413165182439, 26.18946497433581420460487820716, 27.60608655499288959489108899712, 29.07020948143950476309348722912

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