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Properties

Label 1-68-68.63-r0-0-0
Degree $1$
Conductor $68$
Sign $0.151 + 0.988i$
Analytic cond. $0.315790$
Root an. cond. $0.315790$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(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)  = 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

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

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.151 + 0.988i$
Analytic conductor: \(0.315790\)
Root analytic conductor: \(0.315790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (0:\ ),\ 0.151 + 0.988i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 \)
good3 \( 1 + (-0.382 + 0.923i)T \)
5 \( 1 + (0.923 + 0.382i)T \)
7 \( 1 + (-0.923 + 0.382i)T \)
11 \( 1 + (0.382 + 0.923i)T \)
13 \( 1 + iT \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 + (-0.382 - 0.923i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (0.382 - 0.923i)T \)
41 \( 1 + (0.923 - 0.382i)T \)
43 \( 1 + (0.707 + 0.707i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.382 + 0.923i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.382 + 0.923i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.923 - 0.382i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

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