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Properties

Label 1-192-192.11-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: $\chi_{192} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 192,\ (0:\ ),\ -0.290 - 0.956i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5303819462 - 0.7151372689i\)
\(L(\frac12)\) \(\approx\) \(0.5303819462 - 0.7151372689i\)
\(L(1)\) \(\approx\) \(0.8463451949 - 0.3544481097i\)
\(L(1)\) \(\approx\) \(0.8463451949 - 0.3544481097i\)

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

−27.168733626435994738853131685219, −26.28141726574186955964182275548, −25.573249129496534206422221020779, −24.69051166081272876438073010147, −23.33508223973957896868192892416, −22.59004776733778042155700705766, −21.694188390877207970484123070473, −20.85268866020085854131669267098, −19.46929468809421444543548308090, −18.574025969944339983152636121164, −18.03223745011646456338801192482, −16.580905393490824304984514849264, −15.67644652932012394792865743930, −14.639400985109226285481325335287, −13.73551757649785621149535396684, −12.53855480847548067259243399771, −11.573807047873093791557454054435, −10.20930632168181511530190459403, −9.623385305134219122923182347, −8.17436341071652488388174214976, −6.866284136156477506271405388324, −6.07137204800882924618450189264, −4.69441526440908099848630919624, −3.023762521081021201178758266697, −2.206534069886428760627773720, 0.66919504818735222427914933380, 2.47516038337587759875211054161, 3.930100827032720715665927685141, 5.19562028423402198194140661687, 6.21717314830131668555468451939, 7.66623937505928770101558329981, 8.63131839133551402538422395661, 9.9237893460683818799474758707, 10.60159662235410142079029223987, 12.20627892115932444254337379755, 13.14017181769492152744789101455, 13.66958438923431494806050202312, 15.30580418225195942829038704847, 16.12459757593672729028914736629, 17.15573057462929530572369095660, 17.845107283317381717537781339113, 19.42158351212444202090321248989, 19.96083611947871915218937107885, 21.090393693444901759028858893716, 21.87151315368139304800734740752, 23.190633599926566400034938648956, 23.87558600903000306428671405668, 24.890143188090525924735175994907, 25.88190773074425988436391332648, 26.63072238779347803837591281006

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