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Properties

Label 1-116-116.51-r1-0-0
Degree $1$
Conductor $116$
Sign $-0.310 - 0.950i$
Analytic cond. $12.4659$
Root an. cond. $12.4659$
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)3-s + (−0.900 + 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s − 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)21-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)27-s + (−0.900 + 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)3-s + (−0.900 + 0.433i)5-s + (−0.623 − 0.781i)7-s + (−0.222 + 0.974i)9-s + (−0.222 − 0.974i)11-s + (−0.222 − 0.974i)13-s + (−0.900 − 0.433i)15-s − 17-s + (0.623 − 0.781i)19-s + (0.222 − 0.974i)21-s + (0.900 + 0.433i)23-s + (0.623 − 0.781i)25-s + (−0.900 + 0.433i)27-s + (−0.900 + 0.433i)31-s + (0.623 − 0.781i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(12.4659\)
Root analytic conductor: \(12.4659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (1:\ ),\ -0.310 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3472940677 - 0.4789682607i\)
\(L(\frac12)\) \(\approx\) \(0.3472940677 - 0.4789682607i\)
\(L(1)\) \(\approx\) \(0.8241761964 + 0.03736267892i\)
\(L(1)\) \(\approx\) \(0.8241761964 + 0.03736267892i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 \)
good3 \( 1 + (0.623 + 0.781i)T \)
5 \( 1 + (-0.900 + 0.433i)T \)
7 \( 1 + (-0.623 - 0.781i)T \)
11 \( 1 + (-0.222 - 0.974i)T \)
13 \( 1 + (-0.222 - 0.974i)T \)
17 \( 1 - T \)
19 \( 1 + (0.623 - 0.781i)T \)
23 \( 1 + (0.900 + 0.433i)T \)
31 \( 1 + (-0.900 + 0.433i)T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 - T \)
43 \( 1 + (-0.900 - 0.433i)T \)
47 \( 1 + (-0.222 - 0.974i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 - T \)
61 \( 1 + (-0.623 - 0.781i)T \)
67 \( 1 + (0.222 - 0.974i)T \)
71 \( 1 + (0.222 + 0.974i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (-0.222 + 0.974i)T \)
83 \( 1 + (-0.623 + 0.781i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 + (-0.623 + 0.781i)T \)
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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

−29.02673236964777028303300920649, −28.61458787814843831697522660906, −27.17370462916231997370011156486, −26.141094189501026243305813737133, −25.150608378938258014463321694931, −24.29960557480285602374874039217, −23.35918890842910229247297793484, −22.322047284933026905923039744528, −20.75830748419909248039908482264, −19.93549192970869269386077890260, −19.00384975752735099627996382698, −18.253591619717378990531908418667, −16.737641539373155897955781870123, −15.51200920085857680956160963985, −14.72175473453386521539831666306, −13.211964814792954210197330274100, −12.40063992638583393345544147822, −11.569003438240850132829372483842, −9.548035246684427855798826222183, −8.69081020556594289204518808160, −7.49318046573369210773882759757, −6.490981537620674502000702687213, −4.67770370902070374115940603541, −3.20024183239200755387506856500, −1.78946448208466202230015421962, 0.21994717479969991399289956928, 2.97192005447728523564761946414, 3.665885424482630632251590190647, 5.08407275344337047787550092376, 6.94605157407818084975795281286, 8.0257575518852407006940375393, 9.20663641176627155820318897662, 10.56579193947465528521948935637, 11.18465062301931133429171241561, 13.02078099225303841178190032653, 13.96836878699544527958419464035, 15.278589629020330116506356265753, 15.86554579168447122077083112002, 16.96894864859131450142402215915, 18.56987273717288695109990974911, 19.88664769246738571531492331721, 19.98463009604897336699562544478, 21.62892709911352385163217080039, 22.46721097134394051105960930613, 23.45771895285679084865631867106, 24.7115783334603054271701684918, 25.97203388104659192448783803669, 26.822237856701110853022986812710, 27.202667933704949431970164217479, 28.60672676505921829028145979524

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