Properties

Degree $1$
Conductor $1157$
Sign $-0.779 + 0.625i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (0.540 − 0.841i)2-s + (0.936 − 0.349i)3-s + (−0.415 − 0.909i)4-s + (0.142 − 0.989i)5-s + (0.212 − 0.977i)6-s + (−0.599 − 0.800i)7-s + (−0.989 − 0.142i)8-s + (0.755 − 0.654i)9-s + (−0.755 − 0.654i)10-s + (−0.989 + 0.142i)11-s + (−0.707 − 0.707i)12-s + (−0.997 + 0.0713i)14-s + (−0.212 − 0.977i)15-s + (−0.654 + 0.755i)16-s + (0.540 + 0.841i)17-s + (−0.142 − 0.989i)18-s + ⋯
L(s,χ)  = 1  + (0.540 − 0.841i)2-s + (0.936 − 0.349i)3-s + (−0.415 − 0.909i)4-s + (0.142 − 0.989i)5-s + (0.212 − 0.977i)6-s + (−0.599 − 0.800i)7-s + (−0.989 − 0.142i)8-s + (0.755 − 0.654i)9-s + (−0.755 − 0.654i)10-s + (−0.989 + 0.142i)11-s + (−0.707 − 0.707i)12-s + (−0.997 + 0.0713i)14-s + (−0.212 − 0.977i)15-s + (−0.654 + 0.755i)16-s + (0.540 + 0.841i)17-s + (−0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (-0.779 + 0.625i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (-0.779 + 0.625i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1157\)    =    \(13 \cdot 89\)
Sign: $-0.779 + 0.625i$
Motivic weight: \(0\)
Character: $\chi_{1157} (330, \cdot )$
Sato-Tate group: $\mu(88)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1157,\ (0:\ ),\ -0.779 + 0.625i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(-0.6519907409 - 1.854380485i\)
\(L(\frac12,\chi)\) \(\approx\) \(-0.6519907409 - 1.854380485i\)
\(L(\chi,1)\) \(\approx\) \(0.7972120280 - 1.287492039i\)
\(L(1,\chi)\) \(\approx\) \(0.7972120280 - 1.287492039i\)

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

−21.87899238213630117963408524766, −21.06286923139599637516510692403, −20.560864098902909779795434613857, −19.196667476092715975864009209379, −18.47870155721937105225162911527, −18.25137233121323553477265942272, −16.760595615660676755110184686593, −16.02366629307974728777683982769, −15.46485085684781579636591166464, −14.76506222955907919693358441608, −14.14842028354021735280785909132, −13.43708337210785081090018502486, −12.636205960760970082063558409894, −11.75737576659534573542018132254, −10.41303896975629172110800928581, −9.81493552635822357792076196385, −8.83112267030333629321202215497, −8.06651373253765822034738220410, −7.327060949992746660962260969, −6.45903700124005246464068748047, −5.572933936819038740074699648378, −4.71241122770824703321910826747, −3.4136775026910131288571725141, −3.02778556347161756715160278369, −2.21890850394501297753891299211, 0.554340981886660771445723077687, 1.56569525346971708285237493967, 2.47439823017510298123214003979, 3.482317859068747076156128326872, 4.14623234654059912578201568098, 5.11149613226884221914287884624, 6.06522897456880238891866495192, 7.23162874786076842182198071433, 8.12358847077080255748589209616, 9.02885510056266175463978366865, 9.79667550339482309598585327963, 10.332344498398172286790731911, 11.514957709131268971289258299010, 12.60805662661652516137727967857, 12.938648437058353908996162970700, 13.616243734811878007344864126092, 14.14472889181291092707468905279, 15.48714560316761006268231261062, 15.701365116075136618660418067915, 17.161355230280018299221354822540, 17.79219278701389253524256831911, 19.10306379788515060625388538070, 19.29625269070584284839811648527, 20.18481656092088587479380275981, 20.76012371967199594987184545796

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