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Properties

Label 1-1176-1176.251-r1-0-0
Degree $1$
Conductor $1176$
Sign $0.995 + 0.0960i$
Analytic cond. $126.378$
Root an. cond. $126.378$
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  + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)17-s − 19-s + (−0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.900 − 0.433i)29-s + 31-s + (0.900 + 0.433i)37-s + (0.623 + 0.781i)41-s + (0.623 − 0.781i)43-s + (0.222 + 0.974i)47-s + (−0.900 + 0.433i)53-s + (0.623 − 0.781i)55-s + ⋯
L(s)  = 1  + (−0.623 − 0.781i)5-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)17-s − 19-s + (−0.900 + 0.433i)23-s + (−0.222 + 0.974i)25-s + (−0.900 − 0.433i)29-s + 31-s + (0.900 + 0.433i)37-s + (0.623 + 0.781i)41-s + (0.623 − 0.781i)43-s + (0.222 + 0.974i)47-s + (−0.900 + 0.433i)53-s + (0.623 − 0.781i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.995 + 0.0960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $0.995 + 0.0960i$
Analytic conductor: \(126.378\)
Root analytic conductor: \(126.378\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (1:\ ),\ 0.995 + 0.0960i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.140103100 + 0.05486483686i\)
\(L(\frac12)\) \(\approx\) \(1.140103100 + 0.05486483686i\)
\(L(1)\) \(\approx\) \(0.8299623035 - 0.08670606701i\)
\(L(1)\) \(\approx\) \(0.8299623035 - 0.08670606701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (0.222 + 0.974i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.900 - 0.433i)T \)
31 \( 1 + T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (0.623 + 0.781i)T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.222 + 0.974i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.222 - 0.974i)T \)
79 \( 1 - T \)
83 \( 1 + (-0.222 + 0.974i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−21.24002619885222013416847929891, −20.064037388203892062453317706113, −19.36577728363102314540545767253, −18.8885278393902586475510656208, −18.08314841315142494253684412076, −17.1400683778467655652292538625, −16.331205708922502935397460926341, −15.65427353853752651277285590570, −14.7063082575228184229809971887, −14.221239346209624694827842627864, −13.293725813530672702520430935241, −12.33111517920124564562860147662, −11.418934799534234369099987681677, −10.98029005279165801413417179710, −10.09629494519758720033915105942, −8.96333619030317882631560906014, −8.32745237853750518339764906849, −7.34909496947520051918960043974, −6.49483807757861460584914261690, −5.92675787259644655733292061127, −4.3722379013837062362455123442, −3.957089177956143311891042100267, −2.78623436140844595995634481996, −1.9300005976962476573220127477, −0.39649234536594042138110293019, 0.552950805159690957483420617924, 1.76659719500955320081530367358, 2.79323987767044056015502166143, 4.13129550382879274047316136306, 4.53751660960111054520095718459, 5.59208323670643863012466335589, 6.58462421205580072850835258738, 7.66023290844367086173946499402, 8.13851931539702887150925124660, 9.206602424806860057245180109904, 9.837317008912516915697538366658, 10.92257258875389657021550342446, 11.69877780831827747099465816301, 12.58511605373984814557532434679, 12.986065188444732611344368280932, 14.037318894966322982074707933711, 15.21466928333831832853545652116, 15.428606814062727363215510857, 16.40503166796534679708684927135, 17.3470871214341022282834663420, 17.72835919251486227562819777651, 18.88759475020164694471642511973, 19.68226219488148075106182679946, 20.294602753450847012537825322055, 20.77284949220044978177249488359

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