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Properties

Label 1-68-68.23-r0-0-0
Degree $1$
Conductor $68$
Sign $0.563 - 0.825i$
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  + (−0.923 + 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s − i·13-s + (0.707 + 0.707i)15-s + (−0.707 − 0.707i)19-s + i·21-s + (−0.923 − 0.382i)23-s + (−0.707 + 0.707i)25-s + (−0.382 + 0.923i)27-s + (0.382 + 0.923i)29-s + (0.923 − 0.382i)31-s − 33-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)3-s + (−0.382 − 0.923i)5-s + (0.382 − 0.923i)7-s + (0.707 − 0.707i)9-s + (0.923 + 0.382i)11-s − i·13-s + (0.707 + 0.707i)15-s + (−0.707 − 0.707i)19-s + i·21-s + (−0.923 − 0.382i)23-s + (−0.707 + 0.707i)25-s + (−0.382 + 0.923i)27-s + (0.382 + 0.923i)29-s + (0.923 − 0.382i)31-s − 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.563 - 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.563 - 0.825i$
Analytic conductor: \(0.315790\)
Root analytic conductor: \(0.315790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (0:\ ),\ 0.563 - 0.825i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5999161106 - 0.3167724552i\)
\(L(\frac12)\) \(\approx\) \(0.5999161106 - 0.3167724552i\)
\(L(1)\) \(\approx\) \(0.7634939961 - 0.1743218606i\)
\(L(1)\) \(\approx\) \(0.7634939961 - 0.1743218606i\)

Euler product

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

−32.006742524271256679466997661050, −30.74379507162382070360183338192, −30.01129138495850944585071010149, −28.84025084989985765565796069710, −27.76284809159915574785553760327, −26.90339443168161410417611023540, −25.37154978158049417050143823412, −24.26721293039846799833467489602, −23.23359996544690134921882879113, −22.1428724661619247156271228331, −21.44843556606878448519782605815, −19.30996333720959441286853710448, −18.67725712306652090378209668609, −17.57137628746798747257506650160, −16.32417008143078793192716739709, −15.04060507143325823254688667952, −13.82948040852497184076882424638, −11.927075318067764362083577591467, −11.61805868193844080363584040159, −10.142117406337500703107015701249, −8.348381704926303033742887583559, −6.81098451987988478197607103349, −5.900972654192885020608355047147, −4.12535607219326723849643624426, −2.032284337879412991691014166955, 1.00949864740286362917688993190, 4.03048791526852955023295178451, 4.911710674717022426857490336656, 6.52284935105096765866954581694, 8.055276712600028000242244915807, 9.642050176771767665814328476765, 10.884343142288191372533076934998, 12.026017435307007061622917626780, 13.10548384395441569641698853059, 14.83792052729739545308916774212, 16.11281888379317760067207196778, 17.067549428571425093892383814827, 17.814913121739401108255252258211, 19.76664640067378519870353670968, 20.54582083291724124183242001644, 21.85759351885856129654563345897, 23.06294675594861128132176356226, 23.83575716955401811159068884090, 24.94904135053629571004711689284, 26.63982610204069787996333892742, 27.675524483279024416326310191003, 28.1717556344329197922802128130, 29.61171712467465957546007349427, 30.423447843766143327898272574051, 32.194158119702833540010627050766

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