Properties

Degree $1$
Conductor $68$
Sign $0.151 - 0.988i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Related objects

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Normalization:  

Dirichlet series

L(χ,s)  = 1  + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s i·13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)19-s + i·21-s + (−0.382 + 0.923i)23-s + (0.707 − 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.923 + 0.382i)29-s + (0.382 + 0.923i)31-s − 33-s + ⋯
L(s,χ)  = 1  + (−0.382 − 0.923i)3-s + (0.923 − 0.382i)5-s + (−0.923 − 0.382i)7-s + (−0.707 + 0.707i)9-s + (0.382 − 0.923i)11-s i·13-s + (−0.707 − 0.707i)15-s + (0.707 + 0.707i)19-s + i·21-s + (−0.382 + 0.923i)23-s + (0.707 − 0.707i)25-s + (0.923 + 0.382i)27-s + (−0.923 + 0.382i)29-s + (0.382 + 0.923i)31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(\chi,s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(\chi,s)\cr =\mathstrut & (0.151 - 0.988i)\, \Lambda(\overline{\chi},1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s,\chi)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s,\chi)\cr =\mathstrut & (0.151 - 0.988i)\, \Lambda(1-s,\overline{\chi}) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.151 - 0.988i$
Motivic weight: \(0\)
Character: $\chi_{68} (27, \cdot )$
Sato-Tate group: $\mu(16)$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (0:\ ),\ 0.151 - 0.988i)\)

Particular Values

\(L(\chi,\frac{1}{2})\) \(\approx\) \(0.6478012267 - 0.5560125721i\)
\(L(\frac12,\chi)\) \(\approx\) \(0.6478012267 - 0.5560125721i\)
\(L(\chi,1)\) \(\approx\) \(0.8620646216 - 0.3982631844i\)
\(L(1,\chi)\) \(\approx\) \(0.8620646216 - 0.3982631844i\)

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

−32.47283389138975759028571583215, −31.24843738938456445382085554800, −29.809993979513024094304793121049, −28.646479229293526603568462918530, −28.15695560230549670526673877062, −26.37056905652097320789236698680, −26.00427518854684380169958668214, −24.634813433088680752023151806531, −22.87397581567017718755055909914, −22.23133278041971123855300928151, −21.29605307270859314101915722783, −20.09252286260939911304807332312, −18.57293317917904502605492438819, −17.37028110971568676956208493782, −16.36937675124274696794408378870, −15.17144366965938574862837510956, −14.03661475342933455522818666474, −12.50638684457000217696248769211, −11.134394569372804770052855927198, −9.70069761017559633566695659524, −9.30416549637135546809713705322, −6.80053642666094412020419274071, −5.75470217597515484262362592480, −4.20775425633459956630059228287, −2.50822325962434919927242221689, 1.21256573519733457692217203738, 3.12088639999970769558232313377, 5.51769079212955994020735975917, 6.341732347188707446727123203837, 7.84243656393263407006162122968, 9.38514248613902203299747864325, 10.72436905137983553912868828419, 12.29096584894386513025981571848, 13.271437595523997609778218784021, 14.05557323262944999275292509280, 16.14909218229496937713680239356, 17.08311815344879953899252673597, 18.10299928597700969106392143217, 19.29798119122741077006243333311, 20.36048694361486474817545799846, 21.94790593242770874192318624814, 22.80990217024089429690788556973, 24.11916111431913132643969485681, 25.03597049033551915683813470384, 25.87913331124624835763347511923, 27.47478579506427722589227155939, 28.85357840785085612966895844697, 29.431085604377652844860287224955, 30.21838078632377382643994150301, 31.8264516051641793677904399518

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