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Properties

Label 1-168-168.5-r0-0-0
Degree $1$
Conductor $168$
Sign $0.605 + 0.795i$
Analytic cond. $0.780188$
Root an. cond. $0.780188$
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  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + 41-s − 43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)11-s + 13-s + (−0.5 + 0.866i)17-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s + 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)37-s + 41-s − 43-s + (−0.5 − 0.866i)47-s + (−0.5 + 0.866i)53-s − 55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(168\)    =    \(2^{3} \cdot 3 \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.780188\)
Root analytic conductor: \(0.780188\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{168} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 168,\ (0:\ ),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.063078436 + 0.5269969551i\)
\(L(\frac12)\) \(\approx\) \(1.063078436 + 0.5269969551i\)
\(L(1)\) \(\approx\) \(1.086700037 + 0.2627917963i\)
\(L(1)\) \(\approx\) \(1.086700037 + 0.2627917963i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 - T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - T \)
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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

−27.46841557908769473908233916736, −26.56648174426356427545328826644, −25.34542715442905220390303239288, −24.699915642036782105068841133799, −23.66218109044349634941153534067, −22.72773294376656175195887791758, −21.28148282536930433692004795962, −20.93136093995145845963687591194, −19.739657805981922847830843410445, −18.5519593610430566570268363979, −17.678902698656761708163382371668, −16.3930407624520453105724068235, −15.948829636998399013070676765787, −14.32053283916103572861851753651, −13.419578238534493368967336889648, −12.56781693932310750256461757712, −11.26362442819049347614770639276, −10.18912230790625503969547474814, −8.87522574516523765046671825911, −8.218645817454407277565249162923, −6.506974546877272575777273423439, −5.494849880141567620361889156430, −4.30589465701637578992981686514, −2.724348470162724943442533169353, −1.090067819905576783544081608951, 1.840148496902979505368750533542, 3.09015000862237009796125226549, 4.55465340196274107366830350908, 6.0301261806545209355356263346, 6.90864507485057617642562914794, 8.204170231460094678492411088143, 9.537712635710155416437290858543, 10.552922089318167346507614424112, 11.39999495395348731316781174577, 12.94565265843767994374544834745, 13.66931752347756294250965382219, 15.00416410544903206749999889995, 15.57952007996013799984245040066, 17.18901626077457641975544466874, 17.900983743860645599432850874146, 18.83814066906960014029510511752, 19.92251580937376942562405360625, 21.12809869714903857729933876371, 21.85474408297508242512446374442, 23.00297205144492938354087212548, 23.67013931797687740423580482647, 25.13013939299824409244963808982, 25.87582055752188648715974741302, 26.53221835810677635733917341961, 27.88971620729822173395877812473

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