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Properties

Label 1-68-68.31-r0-0-0
Degree $1$
Conductor $68$
Sign $0.990 - 0.139i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 68 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(68\)    =    \(2^{2} \cdot 17\)
Sign: $0.990 - 0.139i$
Analytic conductor: \(0.315790\)
Root analytic conductor: \(0.315790\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{68} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 68,\ (0:\ ),\ 0.990 - 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.186431766 - 0.08301146307i\)
\(L(\frac12)\) \(\approx\) \(1.186431766 - 0.08301146307i\)
\(L(1)\) \(\approx\) \(1.266858080 - 0.04481115819i\)
\(L(1)\) \(\approx\) \(1.266858080 - 0.04481115819i\)

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

−31.812840524550624832792515064065, −30.91268124750830569480694434498, −29.88428850035032853174632863870, −28.97588124688417959307656389144, −27.40764176843798867535522072808, −26.14831334277807088889168347054, −25.54677068205026399008746057718, −24.52837094670946988389707886219, −23.10819784487299049686147326064, −21.82923560608968384525742716179, −20.904338430854355451347333911317, −19.38870712997565186755908824165, −18.643827471163087233822410374692, −17.68690319678594267803076467969, −15.56872149141232608401714135684, −14.96322092719022958293727649985, −13.55063079922134479825155003383, −12.68330982138315305169948149913, −10.89147544192250539013510834210, −9.58390750890875960194823559635, −8.32310002757336929974161013776, −7.00822033934335875904075077332, −5.62610539689233642788624556475, −3.22509841415370874762253247956, −2.369554972275987239337818312421, 1.93240083009867725939841402286, 3.80466797672796764801784803227, 5.006295085566735332340316001584, 7.07222121326173318983131689460, 8.44578506814807660121756211206, 9.55602340663997763471099439333, 10.60507814962296583569553924244, 12.71229317369753158893052350061, 13.4999909673281063675544923668, 14.68927304609614791484781787120, 16.16301527138405442173579819010, 16.88919882499021318299191546543, 18.63366000336412758404040675941, 19.887461641270633754168845580911, 20.72122329942169699607737924979, 21.5111277595416073733155780746, 23.23385937639716677090324936923, 24.295093733333689902480451238586, 25.534972299073245851009669397708, 26.271298745026114039455977405015, 27.41825685552648495305904977405, 28.69118805901459611811273902350, 29.682090066591258197550520293839, 31.16692603538614796850540816802, 31.8285636999183897498996958127

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