↑

Properties

Label 1-2e7-128.115-r1-0-0
Degree $1$
Conductor $128$
Sign $0.740 + 0.671i$
Analytic cond. $13.7555$
Root an. cond. $13.7555$
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

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

Invariants

Degree: \(1\)
Conductor: \(128\)    =    \(2^{7}\)
Sign: $0.740 + 0.671i$
Analytic conductor: \(13.7555\)
Root analytic conductor: \(13.7555\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{128} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 128,\ (1:\ ),\ 0.740 + 0.671i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.859461483 + 0.7172734446i\)
\(L(\frac12)\) \(\approx\) \(1.859461483 + 0.7172734446i\)
\(L(1)\) \(\approx\) \(1.299603485 + 0.1269120957i\)
\(L(1)\) \(\approx\) \(1.299603485 + 0.1269120957i\)

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

−28.049134601234744388233938708643, −27.30515701528840028695259504838, −26.53868666263164010580097432899, −25.6772025897474363162751485985, −24.290461492303693467718528990680, −23.55130411184094463688338880759, −22.40497513320600175328911216290, −20.89245322567609594295915394342, −20.54335353988722390339442866390, −19.388269171964244067336659116852, −18.52880654598114201307070320490, −16.785200590879334314305061809465, −15.99379133641986196156626970086, −15.06228199984451982081361109935, −13.89460543007085912418716848657, −13.04334519163194457625530033130, −11.253139043040760099070324695092, −10.63045969969620870705546290685, −9.04142959280568186425505389973, −8.116032282323390842921506238461, −7.21785259105441109548030945390, −5.11459061155310049052440215354, −3.971240065039230971856868513711, −3.00711008095252987390331181267, −0.80747442867478498704147248432, 1.526096140404063534137762538655, 2.95494529512552581617082382216, 4.17834588294030414258808425940, 5.945582370783524995422707304887, 7.4555269477042229705854951188, 8.17483473433871835151456113382, 9.22670094904705031033122674127, 10.83685873008197846820688413034, 12.15455097487550382645603833252, 12.80931453547901122540753674756, 14.36110949996552655018658207358, 15.11802242345132399629949039930, 15.9582981127099852834017830473, 17.73943582678478806146646306285, 18.74533252055283330394593476635, 19.25731853727678386342479171338, 20.609306224707225654154245804961, 21.246536239206711472360778624701, 23.00012894510497055086847094770, 23.58221438887584351097180800950, 24.7909807420742366621532004487, 25.590847752136243124880144873943, 26.53691491985462127566688973883, 27.69303240077592897191986114865, 28.5082790833243769871896176556

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