L-function 1-4140-4140.1499-r0-0-0

• Label: 1-4140-4140.1499-r0-0-0
• Degree: $1$
• Conductor: $4140$
• Sign: $0.452 + 0.891i$
• Analytic cond.: $19.2260$
• Root an. cond.: $19.2260$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.235 − 0.971i)7^-s + (0.981 − 0.189i)11^-s + (−0.235 − 0.971i)13^-s + (−0.142 + 0.989i)17^-s + (0.142 + 0.989i)19^-s + (0.786 + 0.618i)29^-s + (−0.0475 + 0.998i)31^-s + (−0.415 + 0.909i)37^-s + (−0.580 + 0.814i)41^-s + (0.0475 + 0.998i)43^-s + (0.5 − 0.866i)47^-s + (−0.888 − 0.458i)49^-s + (−0.959 + 0.281i)53^-s + (0.235 + 0.971i)59^-s + (−0.888 + 0.458i)61^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.452 + 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$4140$    =    $2^{2} \cdot 3^{2} \cdot 5 \cdot 23$
Sign:	$0.452 + 0.891i$
Analytic conductor:	$19.2260$
Root analytic conductor:	$19.2260$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{4140} (1499, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 4140,\ (0:\ ),\ 0.452 + 0.891i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$1.276312637 + 0.7838063899i$
$L(1)$	$\approx$	$1.086526073 + 0.03984789521i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	5	$1$
	23	$1$
good	7	$1 + (0.235 - 0.971i)T$
	11	$1 + (0.981 - 0.189i)T$
	13	$1 + (-0.235 - 0.971i)T$
	17	$1 + (-0.142 + 0.989i)T$
	19	$1 + (0.142 + 0.989i)T$
	29	$1 + (0.786 + 0.618i)T$
	31	$1 + (-0.0475 + 0.998i)T$
	37	$1 + (-0.415 + 0.909i)T$
	41	$1 + (-0.580 + 0.814i)T$

Imaginary part of the first few zeros on the critical line

−18.36318264451610880363331900369, −17.49943338973940808113551118077, −17.13785623468506919137323638800, −16.13434851110976604983562635952, −15.61071610122695426027317986786, −14.96135221090334305802603573658, −14.083177148338676511055488810276, −13.81419008938953098008693652420, −12.68198835606972082519150543303, −12.02587619878872739420419854838, −11.56292211323144381855716181784, −10.95936934810370281957639976531, −9.77505897623810473826923568425, −9.2046038858186683368360049346, −8.86012905413665500690567402974, −7.84100862052483346730248034277, −7.00167020438468934927318039191, −6.45938519603067172163931640080, −5.5919804539826234041221609084, −4.779588610014885557326424330234, −4.20923509785882775673818912033, −3.15550726449791767278004112315, −2.30251977223431195130074406722, −1.720331980772165440144520559572, −0.42893766534066000426094692693, 1.17186934969097970375708322046, 1.46826259202924900119612948461, 2.88811457537181326895813042521, 3.57112129719807060566792193555, 4.257197965633711551417314376910, 5.05259501088911317002998733255, 5.9584263747934967972009550120, 6.65533929090488701831878944141, 7.342887364262120968367615322934, 8.24489739411069971120002796300, 8.59218471134142895247771882467, 9.80065433172713245206260746955, 10.26261252913858106929053697228, 10.87557945092199200308850142171, 11.71182596359952758084170964234, 12.42456413460181637992486165690, 13.06021519657474336076790196352, 13.89486370629278589544412271948, 14.43949509854969481664143236299, 15.00406262943093194791199102360, 15.895455507133109406584820518048, 16.65959765212562365756271027206, 17.16951555168700342680013405065, 17.70728283032243151673033203135, 18.49332180766589873701774843263

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