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Properties

Label 1-115-115.103-r0-0-0
Degree $1$
Conductor $115$
Sign $-0.871 - 0.489i$
Analytic cond. $0.534057$
Root an. cond. $0.534057$
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  + (−0.909 − 0.415i)2-s + (0.281 − 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (−0.989 − 0.142i)7-s + (−0.281 − 0.959i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)11-s + (0.909 − 0.415i)12-s + (0.989 − 0.142i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.755 − 0.654i)17-s + (0.540 + 0.841i)18-s + (−0.654 − 0.755i)19-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)2-s + (0.281 − 0.959i)3-s + (0.654 + 0.755i)4-s + (−0.654 + 0.755i)6-s + (−0.989 − 0.142i)7-s + (−0.281 − 0.959i)8-s + (−0.841 − 0.540i)9-s + (−0.415 − 0.909i)11-s + (0.909 − 0.415i)12-s + (0.989 − 0.142i)13-s + (0.841 + 0.540i)14-s + (−0.142 + 0.989i)16-s + (−0.755 − 0.654i)17-s + (0.540 + 0.841i)18-s + (−0.654 − 0.755i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(115\)    =    \(5 \cdot 23\)
Sign: $-0.871 - 0.489i$
Analytic conductor: \(0.534057\)
Root analytic conductor: \(0.534057\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{115} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 115,\ (0:\ ),\ -0.871 - 0.489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1362430788 - 0.5209828663i\)
\(L(\frac12)\) \(\approx\) \(0.1362430788 - 0.5209828663i\)
\(L(1)\) \(\approx\) \(0.4926595344 - 0.3959418248i\)
\(L(1)\) \(\approx\) \(0.4926595344 - 0.3959418248i\)

Euler product

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

−29.23056767351131932241899449283, −28.3837479546086040321468122502, −27.70313865823399473520132755835, −26.50078302300351094267487832590, −25.78518466330640233671208599661, −25.23714728213467880300589379531, −23.57743384188448496867485122194, −22.66607498508630306697751967899, −21.28766491174976525937895300659, −20.26636615182872435353565443014, −19.44754494769057765762590971061, −18.27604505645616893965397527175, −17.05103639313175936800280384234, −16.03851077334163215841091628670, −15.44272885657449773535367239318, −14.33723523612687503957824115764, −12.745692892935406488037680168008, −11.00314042863871769486074458068, −10.18751464917872475321635033421, −9.20416597518506907821126520103, −8.28821346611743955722680685751, −6.73489435166315062243007205841, −5.55540534289610756076629661357, −3.87841263828079817995533620970, −2.242083576916896271682150274499, 0.64228573662456189438573877365, 2.42570353531372195879327927437, 3.48091046473051923794048474737, 6.15464692674351400977029800602, 7.026742193707898355940018446, 8.36093025046632592260954693397, 9.133069171819492767337704420829, 10.64214351053774816309438497014, 11.666906156697963083457104661779, 12.99292806776093839148365510050, 13.56210055835002433891521733971, 15.534561334409634882273767407671, 16.5020292561134521422056431076, 17.74536391377293631465474174559, 18.61480372138170582055617664835, 19.412208242375199074394997084620, 20.22304938952569502978282876205, 21.41534917450999306703323510995, 22.78209619393032891020547082015, 23.98568740199777656488990528763, 25.07172202984292960936573222686, 25.921610042622678794607510239730, 26.61462922578551269595117328972, 28.05771077678479839672496121614, 29.06202121882354543171515607102

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