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Properties

Label 1-115-115.87-r1-0-0
Degree $1$
Conductor $115$
Sign $-0.844 + 0.536i$
Analytic cond. $12.3584$
Root an. cond. $12.3584$
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  + (−0.540 + 0.841i)2-s + (−0.989 + 0.142i)3-s + (−0.415 − 0.909i)4-s + (0.415 − 0.909i)6-s + (−0.755 − 0.654i)7-s + (0.989 + 0.142i)8-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (0.540 + 0.841i)12-s + (−0.755 + 0.654i)13-s + (0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (−0.281 + 0.959i)18-s + (−0.415 − 0.909i)19-s + ⋯
L(s)  = 1  + (−0.540 + 0.841i)2-s + (−0.989 + 0.142i)3-s + (−0.415 − 0.909i)4-s + (0.415 − 0.909i)6-s + (−0.755 − 0.654i)7-s + (0.989 + 0.142i)8-s + (0.959 − 0.281i)9-s + (0.841 − 0.540i)11-s + (0.540 + 0.841i)12-s + (−0.755 + 0.654i)13-s + (0.959 − 0.281i)14-s + (−0.654 + 0.755i)16-s + (0.909 + 0.415i)17-s + (−0.281 + 0.959i)18-s + (−0.415 − 0.909i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.844 + 0.536i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.844 + 0.536i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (87, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ -0.844 + 0.536i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1150589835 + 0.3958394496i\)
\(L(\frac12)\) \(\approx\) \(0.1150589835 + 0.3958394496i\)
\(L(1)\) \(\approx\) \(0.4662932724 + 0.1950329643i\)
\(L(1)\) \(\approx\) \(0.4662932724 + 0.1950329643i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.540 + 0.841i)T \)
3 \( 1 + (-0.989 + 0.142i)T \)
7 \( 1 + (-0.755 - 0.654i)T \)
11 \( 1 + (0.841 - 0.540i)T \)
13 \( 1 + (-0.755 + 0.654i)T \)
17 \( 1 + (0.909 + 0.415i)T \)
19 \( 1 + (-0.415 - 0.909i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (-0.142 + 0.989i)T \)
37 \( 1 + (-0.281 - 0.959i)T \)
41 \( 1 + (-0.959 - 0.281i)T \)
43 \( 1 + (-0.989 + 0.142i)T \)
47 \( 1 + iT \)
53 \( 1 + (0.755 + 0.654i)T \)
59 \( 1 + (0.654 + 0.755i)T \)
61 \( 1 + (-0.142 + 0.989i)T \)
67 \( 1 + (-0.540 + 0.841i)T \)
71 \( 1 + (0.841 + 0.540i)T \)
73 \( 1 + (0.909 - 0.415i)T \)
79 \( 1 + (0.654 + 0.755i)T \)
83 \( 1 + (0.281 + 0.959i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.281 - 0.959i)T \)
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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

−28.57790324091056169830873171492, −27.78218302720857059677007601536, −27.08849971124963242744232898729, −25.57407968040267180068173267586, −24.76748046368274530727742973344, −22.987879677003251153329599317362, −22.45184784483631329617192158703, −21.55808164248920573163632086449, −20.28402354817729446537769529354, −19.076403061125413681713548495169, −18.41969249500625961879904396992, −17.16278181349937994561099323046, −16.5879692976486121897972007207, −15.09906980781854704214219249411, −13.24220605484684404317185335411, −12.21633407440594639250833077746, −11.75358212554844234466275186629, −10.16154789197528352408287091151, −9.6123817257509428195265773369, −7.93223053842143168178783911070, −6.60876445722183836675458033939, −5.16340019103694323127397221448, −3.61936869622254002031326757479, −1.947351664877957806228876941970, −0.28042054885348419812121916720, 1.135788074858887624799662210114, 3.97418681167094799181677004724, 5.30789744454305919136140439258, 6.54786535674862822825468713650, 7.19242043159794253429491526768, 8.99045252552284122384319089329, 10.00935663411354243192075210626, 10.99239520497881340884701300530, 12.41697763211082181518631982326, 13.793991535231860351453303710292, 14.97860912200237111305380309082, 16.37348695990830587389282613560, 16.73071033747133080395793516654, 17.69206955353888428820524892297, 18.99825655747203386106919841532, 19.72996981074026596509838280212, 21.62252241721493941505310533590, 22.515609045591388544302063156535, 23.54001019340102811104544560504, 24.178374521889259951025412798, 25.46588033519461899939848830987, 26.55577301380739896660038021291, 27.30310449712594184399215155350, 28.29876505942169066312552231644, 29.21398657990844800617362118349

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