L-function 1-115-115.103-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	5	$1$
	23	$1$
good	2	$1 + (-0.909 - 0.415i)T$
	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	$1 + (-0.959 + 0.281i)T$

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