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Properties

Label 1-68-68.55-r1-0-0
Degree $1$
Conductor $68$
Sign $0.615 - 0.788i$
Analytic cond. $7.30761$
Root an. cond. $7.30761$
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  + i·3-s − i·5-s − i·7-s − 9-s − i·11-s + 13-s + 15-s + 19-s + 21-s − i·23-s − 25-s − i·27-s − i·29-s + i·31-s + 33-s + ⋯
L(s)  = 1  + i·3-s − i·5-s − i·7-s − 9-s − i·11-s + 13-s + 15-s + 19-s + 21-s − i·23-s − 25-s − i·27-s − i·29-s + i·31-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.615 - 0.788i$
Analytic conductor: \(7.30761\)
Root analytic conductor: \(7.30761\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (1:\ ),\ 0.615 - 0.788i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.293981416 - 0.6313702366i\)
\(L(\frac12)\) \(\approx\) \(1.293981416 - 0.6313702366i\)
\(L(1)\) \(\approx\) \(1.069483852 - 0.1316594787i\)
\(L(1)\) \(\approx\) \(1.069483852 - 0.1316594787i\)

Euler product

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

−31.19559444140688640022615804647, −30.929219950162945460361133657336, −29.749482371121348150403982360519, −28.68493297618495005910273509840, −27.628837697732914860783203067055, −25.852252554606769834446460831213, −25.47151903785208751938072382777, −24.105963986878647741211207094179, −22.94632647562644474479041849255, −22.13586602738546729951655831281, −20.547801650479837540117087844464, −19.18593153970871648123940319231, −18.32204487067672604232720366769, −17.649255210671784825437838825749, −15.7078372595981289059737320025, −14.61753529637072223787744924223, −13.42740525115170186492771092930, −12.12781554313383510668055346607, −11.16176873284845709347399249957, −9.43533293449462584194823355126, −7.90816107576477216381762090213, −6.76103989494615066846100100432, −5.59935294247543643857742433126, −3.17905263225444310197545440854, −1.81608290297430549981426346890, 0.78233919288920358360480950598, 3.475735391769599439548187855580, 4.61269640474115657509676974199, 5.97612465424866988433426872112, 8.08986620244006576139547150554, 9.14930395663210594216591763837, 10.45946514679554343345051515272, 11.54650159754358228805767169968, 13.27788613785467286695970578567, 14.25472847274316575517761318527, 16.05758793563453061646486150187, 16.39406236648201133834654502430, 17.672597350460670690529759709557, 19.52701858731940035128283962589, 20.57580954342557871966573908114, 21.20299893993558589400375836132, 22.642670274743384327920355314854, 23.70617356122254363252477260965, 24.9073489960730760735370232207, 26.34627187893960837758241397193, 27.04404892110194535782280609531, 28.2336936494814445513709244482, 29.03704374903079457094954745912, 30.51375899159975768952073104272, 31.83757255275859575598680726257

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