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Properties

Label 1-115-115.109-r1-0-0
Degree $1$
Conductor $115$
Sign $0.746 - 0.664i$
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  + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.142 − 0.989i)6-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (0.654 − 0.755i)12-s + (0.959 − 0.281i)13-s + (−0.415 − 0.909i)14-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + (−0.415 + 0.909i)18-s + (0.142 − 0.989i)19-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (−0.841 − 0.540i)3-s + (−0.142 + 0.989i)4-s + (−0.142 − 0.989i)6-s + (−0.959 − 0.281i)7-s + (−0.841 + 0.540i)8-s + (0.415 + 0.909i)9-s + (0.654 − 0.755i)11-s + (0.654 − 0.755i)12-s + (0.959 − 0.281i)13-s + (−0.415 − 0.909i)14-s + (−0.959 − 0.281i)16-s + (−0.142 − 0.989i)17-s + (−0.415 + 0.909i)18-s + (0.142 − 0.989i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.746 - 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $0.746 - 0.664i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ 0.746 - 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.224975927 - 0.4661829782i\)
\(L(\frac12)\) \(\approx\) \(1.224975927 - 0.4661829782i\)
\(L(1)\) \(\approx\) \(1.023980081 + 0.08598847539i\)
\(L(1)\) \(\approx\) \(1.023980081 + 0.08598847539i\)

Euler product

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

−29.03009638279592523995142271602, −28.317042810083933528036002865433, −27.62465506971972749823535073592, −26.25109133002141384401510959545, −24.93269320091232641285485374353, −23.48145962792323621122398207434, −22.89805074193550188714083385509, −22.03356724985623646818924029886, −21.169123998537553193151584967101, −20.06330907118872157094421897725, −18.942578593198225424819444217160, −17.84375845860935777781888112667, −16.43176871665979883719796999030, −15.49101601497922806098956910335, −14.390534084008874304271704183228, −12.83930413731739387060737580721, −12.199006415888191910902814544167, −11.008298156461673881436618709689, −10.033145516217154069203886903113, −9.09277214008995646962785458213, −6.56044706201252420423486111610, −5.84494516318612254229191601202, −4.35473671512418420279037088436, −3.42400368573424751273593322957, −1.4153913971700594884414844509, 0.53415434808938891919218619968, 3.03145835158436181933526244046, 4.50496205290606548646168465840, 5.980320287848634049572629417620, 6.56181773932557382883080030107, 7.77377867186651414594674755309, 9.265460336891076125312564828182, 11.07088441033112363880683853184, 12.018470476409130967468468545435, 13.32113401966263137309963260529, 13.71085259356816104484939926534, 15.5663834814475395398675891955, 16.32134487132892435190603201111, 17.208514469748353136101157193095, 18.290688376726286200079542104050, 19.471795750808345612222467995760, 21.00759820352889452122866230913, 22.38585555737529541686071017052, 22.67518792855589760058654092469, 23.80956111081338380572483652610, 24.65278879726830754653870976227, 25.608773950015336621300872357942, 26.71310747539867458087544763164, 27.9245793835919763227235592040, 29.20239327549828170078511876187

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