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Properties

Label 1-1176-1176.437-r0-0-0
Degree $1$
Conductor $1176$
Sign $-0.631 + 0.775i$
Analytic cond. $5.46132$
Root an. cond. $5.46132$
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  + (−0.826 + 0.563i)5-s + (−0.988 − 0.149i)11-s + (0.623 − 0.781i)13-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 + 0.974i)29-s + (0.5 + 0.866i)31-s + (0.733 + 0.680i)37-s + (−0.900 − 0.433i)41-s + (0.900 − 0.433i)43-s + (0.365 + 0.930i)47-s + (−0.733 + 0.680i)53-s + (0.900 − 0.433i)55-s + ⋯
L(s)  = 1  + (−0.826 + 0.563i)5-s + (−0.988 − 0.149i)11-s + (0.623 − 0.781i)13-s + (0.955 − 0.294i)17-s + (−0.5 + 0.866i)19-s + (−0.955 − 0.294i)23-s + (0.365 − 0.930i)25-s + (−0.222 + 0.974i)29-s + (0.5 + 0.866i)31-s + (0.733 + 0.680i)37-s + (−0.900 − 0.433i)41-s + (0.900 − 0.433i)43-s + (0.365 + 0.930i)47-s + (−0.733 + 0.680i)53-s + (0.900 − 0.433i)55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1176 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.631 + 0.775i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1176\)    =    \(2^{3} \cdot 3 \cdot 7^{2}\)
Sign: $-0.631 + 0.775i$
Analytic conductor: \(5.46132\)
Root analytic conductor: \(5.46132\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1176} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1176,\ (0:\ ),\ -0.631 + 0.775i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2596863828 + 0.5466972145i\)
\(L(\frac12)\) \(\approx\) \(0.2596863828 + 0.5466972145i\)
\(L(1)\) \(\approx\) \(0.7634437960 + 0.1412198873i\)
\(L(1)\) \(\approx\) \(0.7634437960 + 0.1412198873i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.826 + 0.563i)T \)
11 \( 1 + (-0.988 - 0.149i)T \)
13 \( 1 + (0.623 - 0.781i)T \)
17 \( 1 + (0.955 - 0.294i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (-0.222 + 0.974i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.733 + 0.680i)T \)
41 \( 1 + (-0.900 - 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (-0.733 + 0.680i)T \)
59 \( 1 + (-0.826 - 0.563i)T \)
61 \( 1 + (-0.733 - 0.680i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (-0.623 - 0.781i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
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

−21.01951357447201291537979652136, −20.20082414292079336212208766467, −19.39132289378582417864215604240, −18.770049361360066383252509941656, −17.971236023526849464565288301620, −16.9226905852783990082751802744, −16.333701270873487197581107347639, −15.535463285857747260823284008573, −15.00022663240765059827973915796, −13.80064434391034896023111394576, −13.1615099124200022509631712730, −12.30247335441019010930405746430, −11.58205911837511908937987046016, −10.840220674624060173734727276096, −9.83028202798584888092603464045, −8.987501001363913329251449361503, −8.0165941301461717051261679427, −7.637683031582291003534032874047, −6.40668048748560038737270586685, −5.51758204484391646306777842045, −4.49617842527051217204959685557, −3.888987622552706068471785790032, −2.73882921692409176038406199285, −1.60137820790081956781204246807, −0.26345856799466813427789743316, 1.225836442853930962937503021452, 2.69236384408102741647594727138, 3.33531725469878084706358043424, 4.259667892856806106491126172863, 5.37831962657369680841511591178, 6.15924184266022097443555974726, 7.230461609324712146971302642684, 8.02580458157330979889663807312, 8.44880966491800483493133928983, 9.9054600420917695108798414837, 10.53971919961841707605300297298, 11.159450959059979988936101905918, 12.273073609032358269693768131065, 12.68840750999466818837506272291, 13.92714018053212449910508081596, 14.474232023059288850216405973454, 15.54436024015338963221006812721, 15.874241592269851503892528685841, 16.77688065374872793585274178171, 17.86281430914092766029177822180, 18.66002260224691613136099743252, 18.894329058484744903286194389352, 20.17433063836926460828225722340, 20.534041362078649267749535686002, 21.54696204228826358743972953172

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