Properties

Degree $1$
Conductor $1157$
Sign $0.518 + 0.855i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.518 + 0.855i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.518 + 0.855i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $0.518 + 0.855i$
Motivic weight: \(0\)
Character: $\chi_{1157} (369, \cdot )$
Sato-Tate group: $\mu(88)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ 0.518 + 0.855i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(2.471223271 + 1.392019036i\)
\(L(\frac12,\chi)\) \(\approx\) \(2.471223271 + 1.392019036i\)
\(L(\chi,1)\) \(\approx\) \(1.850125023 + 0.4031555317i\)
\(L(1,\chi)\) \(\approx\) \(1.850125023 + 0.4031555317i\)

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

−21.175841085066233924454701807874, −20.51215297818496173717422049791, −19.71028876496840882821885080959, −18.82035148780114653182610926712, −17.98645120824542208337404028326, −17.04771972706989509786649928590, −16.55160859806666591519203059167, −15.95779196008701863368232998782, −14.411257039889312669182196663229, −14.000566641952647167611883954, −13.50035923130148530556310552353, −12.64601365696684804361384829768, −12.15460744579886019861240644643, −11.1565970251360908352794204174, −10.123078051693035189837420432082, −9.07524901715380007974973268881, −8.08207772293581950947440577237, −7.2925200333566766358274009996, −6.37908616603687659271873187444, −5.90721846067099912993304536885, −5.21509845282277640660331111624, −3.75858658710881579269847449055, −3.00241785058998268527685553936, −2.03166780733175341447863960506, −0.8609794414667503256488576733, 1.45671702642754145553237299897, 2.62990650501555159252094249554, 3.19061148491524742430011597872, 4.19830726744911377226477554646, 5.114332277554680292128069963457, 5.8937228613983392742690251630, 6.391489158780153941638120882027, 7.615111127770497267141526761942, 9.34428545744418725643590734009, 9.63678396366169798740897517098, 10.20941670962204328889529659194, 11.21681630829357678173325661556, 11.994179842955673783767737312539, 12.85302847387740797933921447317, 13.61240253555282268424504981032, 14.52279173758295039462626227292, 15.0006573711900120995721425824, 15.828723873599197483796353169886, 16.54153506881476911393188912004, 17.44837162524462010873676054049, 18.40382080602098482392534885156, 19.3642373781359513302493786706, 20.17696720983902574598293489291, 20.761411369470900873877166086852, 21.622791803762353604421878551064

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