Properties

Degree 1
Conductor $ 2^{2} \cdot 17 $
Sign $0.151 + 0.988i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

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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

\( d \)  =  \(1\)
\( N \)  =  \(68\)    =    \(2^{2} \cdot 17\)
\( \varepsilon \)  =  $0.151 + 0.988i$
motivic weight  =  \(0\)
character  :  $\chi_{68} (63, \cdot )$
Sato-Tate  :  $\mu(16)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 68,\ (0:\ ),\ 0.151 + 0.988i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.6478012267 + 0.5560125721i$
$L(\frac12,\chi)$  $\approx$  $0.6478012267 + 0.5560125721i$
$L(\chi,1)$  $\approx$  0.8620646216 + 0.3982631844i
$L(1,\chi)$  $\approx$  0.8620646216 + 0.3982631844i

Euler product

\[\begin{aligned} L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]
\[\begin{aligned} L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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