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Properties

Label 1-115-115.3-r1-0-0
Degree $1$
Conductor $115$
Sign $-0.300 - 0.953i$
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  + (0.281 + 0.959i)2-s + (0.755 − 0.654i)3-s + (−0.841 + 0.540i)4-s + (0.841 + 0.540i)6-s + (−0.909 − 0.415i)7-s + (−0.755 − 0.654i)8-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.281 + 0.959i)12-s + (−0.909 + 0.415i)13-s + (0.142 − 0.989i)14-s + (0.415 − 0.909i)16-s + (0.540 − 0.841i)17-s + (0.989 − 0.142i)18-s + (−0.841 + 0.540i)19-s + ⋯
L(s)  = 1  + (0.281 + 0.959i)2-s + (0.755 − 0.654i)3-s + (−0.841 + 0.540i)4-s + (0.841 + 0.540i)6-s + (−0.909 − 0.415i)7-s + (−0.755 − 0.654i)8-s + (0.142 − 0.989i)9-s + (−0.959 − 0.281i)11-s + (−0.281 + 0.959i)12-s + (−0.909 + 0.415i)13-s + (0.142 − 0.989i)14-s + (0.415 − 0.909i)16-s + (0.540 − 0.841i)17-s + (0.989 − 0.142i)18-s + (−0.841 + 0.540i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.300 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.300 - 0.953i$
Analytic conductor: \(12.3584\)
Root analytic conductor: \(12.3584\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (1:\ ),\ -0.300 - 0.953i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4098728619 - 0.5585969550i\)
\(L(\frac12)\) \(\approx\) \(0.4098728619 - 0.5585969550i\)
\(L(1)\) \(\approx\) \(0.9392126136 + 0.08461386273i\)
\(L(1)\) \(\approx\) \(0.9392126136 + 0.08461386273i\)

Euler product

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

−29.35397979159761595724678622053, −28.30017680403326750205872063802, −27.53034795635205928147940960651, −26.30919503612146746910703613626, −25.62085189941271521662202627224, −24.1117987209843667277304561616, −22.840202207337071654267271815989, −21.89099523340548938901349351476, −21.19430929522024363038471368820, −20.02747531517911695629349195311, −19.37356051934716646707412819811, −18.35672382389142746941561031670, −16.76158278425550838794729135755, −15.30958401838329613599404831070, −14.67912123163407785203818887423, −13.14564552594562762053600572555, −12.641838273730668037301338978163, −10.90726650013486896671225508407, −9.98027411976213007842178350523, −9.182705411520856774098314634752, −7.84681478551477540917972440118, −5.715494251856968465732692149240, −4.47422169969439434176578907317, −3.15758070713358036095612906034, −2.26869606032832739691529199833, 0.22390554121599431317014388325, 2.68876876088831000044918288147, 3.95338953463660073117938212064, 5.64640007028111385085380571605, 6.96432603477314954533652740110, 7.66448511272171889248321100044, 8.94824167583951976480632472526, 10.00856194441828093103515375515, 12.23457642735406772388337938559, 13.08957493690968793715517527243, 13.95231854881730831865448262824, 14.94551615992978673632666011881, 16.101273982391942660174480250308, 17.08693125055655494790236078731, 18.45821314645765238588308715414, 19.131650119488305925593956521363, 20.49501080234296327386494067045, 21.67223791572738846795082543060, 23.01783525290992147306815439875, 23.70886890035678038927676642534, 24.741449254376235100416697544694, 25.64208550426462906964444535108, 26.346005809080629131828543558738, 27.2251174313858232339832438984, 29.04338543796135012065822637152

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