Properties

Degree $1$
Conductor $1068$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

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Normalization:  

Dirichlet series

L(χ,s)  = 1  − 5-s + 7-s + 11-s − 13-s − 17-s + 19-s − 23-s + 25-s + 29-s + 31-s − 35-s − 37-s + 41-s + 43-s + 47-s + 49-s − 53-s − 55-s − 59-s − 61-s + 65-s − 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯
L(s,χ)  = 1  − 5-s + 7-s + 11-s − 13-s − 17-s + 19-s − 23-s + 25-s + 29-s + 31-s − 35-s − 37-s + 41-s + 43-s + 47-s + 49-s − 53-s − 55-s − 59-s − 61-s + 65-s − 67-s + 71-s + 73-s + 77-s − 79-s − 83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 1068 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & \, \Lambda(\chi,1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 1068 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & \, \Lambda(1-s,\chi) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1068\)    =    \(2^{2} \cdot 3 \cdot 89\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{1068} (1067, \cdot )$
Sato-Tate group: $\mu(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1068,\ (0:\ ),\ 1)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(1.378907357\)
\(L(\frac12,\chi)\) \(\approx\) \(1.378907357\)
\(L(\chi,1)\) \(\approx\) \(1.037592936\)
\(L(1,\chi)\) \(\approx\) \(1.037592936\)

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

−21.55090844219029528654784453822, −20.54882345408904717005136428401, −19.82926432620817913358284217576, −19.417363952307215234402638764442, −18.32444875820929241517057586038, −17.5540759534340394371082530232, −16.94724442862569173753386276056, −15.77501451140975089574306903941, −15.39324586199129465901791075835, −14.19105803503237525141982355261, −14.06592075330454358156204590681, −12.45003603708809874853314346386, −11.95325784991273390584655771303, −11.328531814643779221961375093412, −10.4495471402286869614757432480, −9.338655209613799579499553899273, −8.539407913400381241626295099967, −7.69885133982395807891951912699, −7.07934215185009627459581934006, −5.98968780931053622146424071192, −4.65361055583696693284746620245, −4.38838303397120434443057926847, −3.17701587015051650335911700677, −2.05135319452725079683725717177, −0.85457366495668451020363577401, 0.85457366495668451020363577401, 2.05135319452725079683725717177, 3.17701587015051650335911700677, 4.38838303397120434443057926847, 4.65361055583696693284746620245, 5.98968780931053622146424071192, 7.07934215185009627459581934006, 7.69885133982395807891951912699, 8.539407913400381241626295099967, 9.338655209613799579499553899273, 10.4495471402286869614757432480, 11.328531814643779221961375093412, 11.95325784991273390584655771303, 12.45003603708809874853314346386, 14.06592075330454358156204590681, 14.19105803503237525141982355261, 15.39324586199129465901791075835, 15.77501451140975089574306903941, 16.94724442862569173753386276056, 17.5540759534340394371082530232, 18.32444875820929241517057586038, 19.417363952307215234402638764442, 19.82926432620817913358284217576, 20.54882345408904717005136428401, 21.55090844219029528654784453822

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