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Properties

Label 1-192-192.131-r0-0-0
Degree $1$
Conductor $192$
Sign $0.290 - 0.956i$
Analytic cond. $0.891644$
Root an. cond. $0.891644$
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)5-s + (−0.707 + 0.707i)7-s + (0.923 − 0.382i)11-s + (0.382 − 0.923i)13-s − i·17-s + (0.382 − 0.923i)19-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (−0.923 − 0.382i)29-s + 31-s + (0.923 + 0.382i)35-s + (−0.382 − 0.923i)37-s + (0.707 + 0.707i)41-s + (0.923 − 0.382i)43-s − i·47-s + ⋯
L(s)  = 1  + (−0.382 − 0.923i)5-s + (−0.707 + 0.707i)7-s + (0.923 − 0.382i)11-s + (0.382 − 0.923i)13-s − i·17-s + (0.382 − 0.923i)19-s + (−0.707 − 0.707i)23-s + (−0.707 + 0.707i)25-s + (−0.923 − 0.382i)29-s + 31-s + (0.923 + 0.382i)35-s + (−0.382 − 0.923i)37-s + (0.707 + 0.707i)41-s + (0.923 − 0.382i)43-s − i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.290 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(192\)    =    \(2^{6} \cdot 3\)
Sign: $0.290 - 0.956i$
Analytic conductor: \(0.891644\)
Root analytic conductor: \(0.891644\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{192} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 192,\ (0:\ ),\ 0.290 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7704197077 - 0.5713821971i\)
\(L(\frac12)\) \(\approx\) \(0.7704197077 - 0.5713821971i\)
\(L(1)\) \(\approx\) \(0.9089399805 - 0.2561186283i\)
\(L(1)\) \(\approx\) \(0.9089399805 - 0.2561186283i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.382 - 0.923i)T \)
7 \( 1 + (-0.707 + 0.707i)T \)
11 \( 1 + (0.923 - 0.382i)T \)
13 \( 1 + (0.382 - 0.923i)T \)
17 \( 1 - iT \)
19 \( 1 + (0.382 - 0.923i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
29 \( 1 + (-0.923 - 0.382i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.382 - 0.923i)T \)
41 \( 1 + (0.707 + 0.707i)T \)
43 \( 1 + (0.923 - 0.382i)T \)
47 \( 1 - iT \)
53 \( 1 + (-0.923 + 0.382i)T \)
59 \( 1 + (0.382 + 0.923i)T \)
61 \( 1 + (-0.923 - 0.382i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (0.707 + 0.707i)T \)
79 \( 1 + iT \)
83 \( 1 + (-0.382 + 0.923i)T \)
89 \( 1 + (0.707 - 0.707i)T \)
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

−27.16792707369892515317074600173, −26.147411713085298299332654022316, −25.74794643785004066163396397515, −24.29453424590591066322567919953, −23.29961988476842615240159737661, −22.59445282479406489577297575224, −21.74946445735226705836623405989, −20.43490182781641802085735656260, −19.41970088682467717272376227420, −18.86170989759774937036690325819, −17.558293195285664745770588613582, −16.61059638851498803761584834493, −15.59168946143793403980894568085, −14.45906300997291761687385583381, −13.75706796901486980976794862909, −12.371394941533524688280627123011, −11.39385659265008978868777335654, −10.323270343464711238677438699064, −9.42826908151735781654801196630, −7.921555537800598264741259744618, −6.85191340942384255611768321513, −6.11854547280180245031037520969, −4.08867786770784993639341791308, −3.49416128349911271833936101621, −1.71255706386065786021717157513, 0.7979712642686732686142777796, 2.7069270335384313977982506703, 3.99643315049191803411315277472, 5.304488657080333626699896844027, 6.33314429177044550565798428114, 7.78131004520313083672210242495, 8.92059585512714649082671863072, 9.56271699032335374240612173296, 11.18752872217067576958413728255, 12.144863905279342745022591629040, 12.954534644516846591815433736468, 14.05031383497374315548809419838, 15.52434171077037933544073228847, 16.028554510433726810835083823483, 17.10504787195451735737319207164, 18.231176997879133574845781465291, 19.36308905979048812678358370433, 20.08374347230971349989176110974, 21.05487166556166709953182512503, 22.313397098136815353652806120866, 22.87001454899104009767308830534, 24.36777935335100117657658346668, 24.7572488318086396519544585642, 25.81384699065526517650376604885, 27.00951079827215540125396640835

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