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Properties

Label 1-1256-1256.13-r0-0-0
Degree $1$
Conductor $1256$
Sign $-0.922 + 0.384i$
Analytic cond. $5.83283$
Root an. cond. $5.83283$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s − 23-s + (−0.5 + 0.866i)25-s − 27-s + 29-s + (−0.5 + 0.866i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s − 7-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 − 0.866i)13-s + (0.5 − 0.866i)15-s + (−0.5 − 0.866i)17-s + (0.5 + 0.866i)19-s + (−0.5 − 0.866i)21-s − 23-s + (−0.5 + 0.866i)25-s − 27-s + 29-s + (−0.5 + 0.866i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1256 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1256 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1256\)    =    \(2^{3} \cdot 157\)
Sign: $-0.922 + 0.384i$
Analytic conductor: \(5.83283\)
Root analytic conductor: \(5.83283\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1256} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1256,\ (0:\ ),\ -0.922 + 0.384i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1187186930 + 0.5929935900i\)
\(L(\frac12)\) \(\approx\) \(0.1187186930 + 0.5929935900i\)
\(L(1)\) \(\approx\) \(0.8268783315 + 0.2418053953i\)
\(L(1)\) \(\approx\) \(0.8268783315 + 0.2418053953i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
157 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 - T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 - T \)
29 \( 1 + T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.5 + 0.866i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.5 + 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.46169494368389577216630575824, −19.55950327842052258938949060436, −19.38617462537191579615782787305, −18.56425193711148154930142942278, −17.996706632161845872899958916122, −16.89656616777886600761823526951, −16.07503147329528846942609366887, −15.26404094902428727303792459748, −14.50551619841021686934626311404, −13.50922694302569722524928852955, −13.39359196613631896840513192098, −11.963543748058845528586246040030, −11.67001930004661014481983873766, −10.62138508046598288485103708135, −9.624505598838757648161682715498, −8.728234644471380375110049939206, −8.10273999582969472319197912567, −6.93037001510939363681039312828, −6.573558250392923757196996106363, −5.901430956047407050056978028894, −4.1050833770871491887508406059, −3.459998734677049467532756378034, −2.70959900971506164077607004225, −1.6496841946221025651092187856, −0.22251073451696206020058259480, 1.37859360126240022568507521985, 2.75033113808867174339267128458, 3.582043508120663979262170813960, 4.2963164847369630353620210618, 5.132497526765673473421501744231, 6.06172267122747152192868178159, 7.25687477409186404884815795543, 8.12767579385021627528427139096, 8.87394039065671208135095370033, 9.67978956002326848489491819509, 10.12112335401849394243790879548, 11.26322610059857627703761992526, 12.19327132689046165093271881144, 12.8309600403964442983587738319, 13.72757377265949163625878868791, 14.529798855105590573950183823541, 15.571321621460232714286651478929, 15.95389222922087905106354114493, 16.470435450575831422910784038590, 17.467256541806650250933761219000, 18.37408040578464320236600460987, 19.522025433559933350362243310556, 20.11331686407669472253003639068, 20.302210555417378609954396518409, 21.29663304022092272981787893474

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