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Properties

Label 1-117-117.31-r1-0-0
Degree $1$
Conductor $117$
Sign $-0.892 - 0.451i$
Analytic cond. $12.5733$
Root an. cond. $12.5733$
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  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + i·8-s − 10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − i·19-s + (−0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.866 − 0.5i)7-s + i·8-s − 10-s + (−0.866 − 0.5i)11-s + (−0.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s − i·19-s + (−0.866 − 0.5i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + (0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.892 - 0.451i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.892 - 0.451i$
Analytic conductor: \(12.5733\)
Root analytic conductor: \(12.5733\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 117,\ (1:\ ),\ -0.892 - 0.451i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1266294037 + 0.5306095349i\)
\(L(\frac12)\) \(\approx\) \(-0.1266294037 + 0.5306095349i\)
\(L(1)\) \(\approx\) \(0.8976870302 + 0.4543403061i\)
\(L(1)\) \(\approx\) \(0.8976870302 + 0.4543403061i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 - T \)
19 \( 1 - iT \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + iT \)
41 \( 1 + (-0.866 + 0.5i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.866 - 0.5i)T \)
53 \( 1 + T \)
59 \( 1 + (0.866 - 0.5i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.866 + 0.5i)T \)
71 \( 1 - iT \)
73 \( 1 + iT \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + iT \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−28.62573406107125759240176606655, −27.71549323728866942849574045679, −26.35217915260420161078309703049, −25.00794753421327309609230371896, −24.148419510587189183483102551135, −23.02131213568001545054071486561, −22.5014684527073267588000298298, −21.10402260628116490007499053253, −20.26881158314236811486940498879, −19.31617965787783524973471259552, −18.42195979625757887319962508810, −16.46342148292252218211441944882, −15.63616387450727089212127008325, −14.80964936698101004091202322522, −13.127747436652598128272126955836, −12.623240152341628279873524100393, −11.53449753716784057009419763687, −10.32209546448783514144534578800, −9.02781710143890221944423879811, −7.438063614967043678260067708, −6.04097140135746001417162964216, −4.77403917257320724619188445101, −3.622785315600182259083961555474, −2.25101026939201521578232489986, −0.15689592710593114809267999383, 2.83687004210265213998978094498, 3.749831747646496417683548841710, 5.12800840282505951105425836141, 6.670396320261653506665161668643, 7.36946323390443955550265591828, 8.72470365324671002050631344958, 10.637832315252162314468616014789, 11.52954933258102186616833095832, 12.93417496189264666304096438386, 13.614435208832089391572524374846, 15.06617378236348300660777121407, 15.77013348211674323067038675389, 16.66194964505552087342016317297, 18.08328653188297580427027374393, 19.46613990200325010776368561226, 20.29360446183267066679185438513, 21.74311346011527757913953915801, 22.49433067649653169219625552845, 23.55723737800632562034096691856, 24.02182427069348002561463900207, 25.60451849386236959103046070749, 26.29726464241893544742552964008, 27.1505479886659659154249282479, 28.85575980958928676204314450867, 29.68163473633696515552997866249

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