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Properties

Label 1-116-116.7-r1-0-0
Degree $1$
Conductor $116$
Sign $0.902 + 0.430i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(116\)    =    \(2^{2} \cdot 29\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(12.4659\)
Root analytic conductor: \(12.4659\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{116} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 116,\ (1:\ ),\ 0.902 + 0.430i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8285247046 + 0.1874148262i\)
\(L(\frac12)\) \(\approx\) \(0.8285247046 + 0.1874148262i\)
\(L(1)\) \(\approx\) \(0.7129452174 - 0.06528220615i\)
\(L(1)\) \(\approx\) \(0.7129452174 - 0.06528220615i\)

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

−28.62088000085811090647563000813, −28.00441996389721649833712489855, −27.01076769680732579441553979503, −26.1738193143566433081183606296, −24.72427306289181899508337983298, −23.66303002569236266626889727795, −22.81278861560877211776647182621, −21.69509147632965495966923441668, −21.00907670093135788778509550540, −19.46364465535499796216725555167, −18.82061006883574386885770270959, −17.12030518250194140419061764439, −16.290980578585981338274776347471, −15.601059677994161237545254766, −14.44196685291258779621391404568, −12.66167058125710601623419946304, −11.80567340540236154529422348640, −10.891279346276198033899021806919, −9.36317107613046085587885404522, −8.60524973684631500605949797901, −6.780996033151392406933559238176, −5.547309932408782706625678726504, −4.32848891142691663309920240519, −3.12294978759021914863552009818, −0.51677921644108559337804193588, 0.98171690985996405941174170495, 3.00062813988840488929398794938, 4.487229011644316619231248598221, 6.11589085897829882494714421582, 7.276144342405137269404297162203, 7.898852450078709607304985180383, 9.98583884996430338964885582042, 10.9306953577522427345975598682, 12.21504179585472952611023730398, 12.8654062001522189267632577370, 14.30099066514272751868437565362, 15.50039550015426829573699209462, 16.74710454346993849529300185989, 17.57668190667016789096324018125, 18.892962938516577538647948614436, 19.51294976872907263675353749548, 20.63394109643200823958996005366, 22.52958987815689831837216904450, 22.93701912283610406582074741128, 23.657842154798201172999690108688, 25.01821779566411291874262096624, 25.895565411955411025421404305507, 27.31082805917381695259576212360, 27.928559988291825664378736719882, 29.31690132984227489789806234320

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