Properties

Degree $1$
Conductor $1157$
Sign $0.627 - 0.778i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

Learn more about

Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.909 + 0.415i)2-s + (0.997 + 0.0713i)3-s + (0.654 − 0.755i)4-s + (0.959 + 0.281i)5-s + (−0.936 + 0.349i)6-s + (−0.479 − 0.877i)7-s + (−0.281 + 0.959i)8-s + (0.989 + 0.142i)9-s + (−0.989 + 0.142i)10-s + (−0.281 − 0.959i)11-s + (0.707 − 0.707i)12-s + (0.800 + 0.599i)14-s + (0.936 + 0.349i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 0.415i)17-s + (−0.959 + 0.281i)18-s + ⋯
L(s,χ)  = 1  + (−0.909 + 0.415i)2-s + (0.997 + 0.0713i)3-s + (0.654 − 0.755i)4-s + (0.959 + 0.281i)5-s + (−0.936 + 0.349i)6-s + (−0.479 − 0.877i)7-s + (−0.281 + 0.959i)8-s + (0.989 + 0.142i)9-s + (−0.989 + 0.142i)10-s + (−0.281 − 0.959i)11-s + (0.707 − 0.707i)12-s + (0.800 + 0.599i)14-s + (0.936 + 0.349i)15-s + (−0.142 − 0.989i)16-s + (−0.909 − 0.415i)17-s + (−0.959 + 0.281i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.627 - 0.778i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.627 - 0.778i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.627 - 0.778i$
Motivic weight: \(0\)
Character: $\chi_{1157} (359, \cdot )$
Sato-Tate group: $\mu(88)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.627 - 0.778i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.319503487 - 0.6314087369i\)
\(L(\frac12,\chi)\) \(\approx\) \(1.319503487 - 0.6314087369i\)
\(L(\chi,1)\) \(\approx\) \(1.072700456 - 0.07123084408i\)
\(L(1,\chi)\) \(\approx\) \(1.072700456 - 0.07123084408i\)

Euler product

   \(L(\chi,s) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)
   \(L(s,\chi) = \displaystyle\prod_p (1- \chi(p) p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−21.33887503830613288421887930425, −20.28473826717057774565616282397, −20.085908414783325929627807285629, −18.945701110421471527920246990390, −18.41361546425300791643006521210, −17.829942896176136420482265602082, −16.83326556965836157090237272855, −16.085931174298775686239706514594, −15.12253829567334218600532167071, −14.61825025757377335443194931380, −13.18811096470917909908342031526, −12.834030234075571573546964256494, −12.2070708036844598396589589996, −10.79069699857837866937190645938, −10.04932810262516361966340343249, −9.41850219714282155454321395895, −8.76296581389013635695509678189, −8.20373568782719711388558353988, −6.95439214713977958440351875391, −6.418007690224871702185880265505, −5.06339149653762175757834348452, −3.86629870623281856622936417614, −2.74128576977511773942704409503, −2.16162097529945802324064628653, −1.46725156046881832918495264987, 0.66714492726856002635216113473, 1.94127231303402406131372497876, 2.66004388007914228720803966049, 3.67240816472859504703416602404, 5.001126017982927304206162852084, 6.106336658034306889577462417722, 6.884690075742084379561696368665, 7.52488568596907377211944518050, 8.53808461230083012806537317057, 9.30856934987660403417767084133, 9.74500183418730072206669191429, 10.75300121344079677036560087782, 11.177928779088376030836508310175, 13.05249037918364004432525667508, 13.48806564618520000199359758978, 14.15937544178350749638929095614, 15.07377930538335699456769076089, 15.7464749728866417359874595564, 16.64902935522868383864145473988, 17.27463902743888157377462074236, 18.16617496838414748980240103458, 18.8718908868310453621301714954, 19.557698183502299216847633895, 20.19688980671544074978945362165, 21.0077039214334532076100904441

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