L-function 1-116-116.63-r1-0-0

• Label: 1-116-116.63-r1-0-0
• Degree: $1$
• Conductor: $116$
• Sign: $-0.727 + 0.686i$
• Analytic cond.: $12.4659$
• Root an. cond.: $12.4659$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.900 + 0.433i)3^-s + (−0.222 + 0.974i)5^-s + (0.900 − 0.433i)7^-s + (0.623 − 0.781i)9^-s + (0.623 + 0.781i)11^-s + (0.623 + 0.781i)13^-s + (−0.222 − 0.974i)15^-s − 17^-s + (−0.900 − 0.433i)19^-s + (−0.623 + 0.781i)21^-s + (0.222 + 0.974i)23^-s + (−0.900 − 0.433i)25^-s + (−0.222 + 0.974i)27^-s + (−0.222 + 0.974i)31^-s + (−0.900 − 0.433i)33^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 116 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.727 + 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$116$    =    $2^{2} \cdot 29$
Sign:	$-0.727 + 0.686i$
Analytic conductor:	$12.4659$
Root analytic conductor:	$12.4659$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{116} (63, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 116,\ (1:\ ),\ -0.727 + 0.686i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3700382744 + 0.9309512216i$
$L(1)$	$\approx$	$0.7418966932 + 0.3609039609i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	29	$1$
good	3	$1 + (-0.900 + 0.433i)T$
	5	$1 + (-0.222 + 0.974i)T$
	7	$1 + (0.900 - 0.433i)T$
	11	$1 + (0.623 + 0.781i)T$
	13	$1 + (0.623 + 0.781i)T$
	17	$1 - T$
	19	$1 + (-0.900 - 0.433i)T$
	23	$1 + (0.222 + 0.974i)T$
	31	$1 + (-0.222 + 0.974i)T$

Imaginary part of the first few zeros on the critical line

−28.42557725885463429179096341464, −27.80806015986273436316260926185, −27.00047639661863025054459155608, −25.06798227099674184757575058805, −24.48411453865095152537222093334, −23.69825931996011584650989854920, −22.553489342757701342313712346565, −21.477956107266114986841466990129, −20.464194851215106279630863634219, −19.17780931777788366158382801959, −18.09370690162167362187760838060, −17.153063569077814418186798308250, −16.302467353683596203040259750006, −15.10505341486239895871098366690, −13.51842914359371765458459814154, −12.566069643734215363437941179245, −11.56303213509923316826855597517, −10.71430556869625067727891474723, −8.81528334635240403417324309182, −8.036574577978951176872075268760, −6.35479402886231889162471954369, −5.33099020864045719247001942250, −4.202510078246022221902240202991, −1.81487301258962716689771888051, −0.48540657349548177258942072483, 1.68532949907392948284878133972, 3.82644700916630757462806847640, 4.766657294874706325755261119307, 6.4438696631553574131692179394, 7.18585860245597687574030948736, 8.967345198660467885533812466486, 10.41617285761140462749461685622, 11.16289680699565123375085437958, 11.97599408588127071195967049795, 13.68422564502889929538093955242, 14.88095297698813281786551597013, 15.661432922315774937483899478924, 17.17440483303410130404001597469, 17.72557048074977710961937979413, 18.86557639296722033663144860108, 20.23231946800144910196048349696, 21.45044519415071394803074774053, 22.19898880395788810549936443229, 23.338822602805868809382400450679, 23.84664692861031135161498832201, 25.48163918126750543435362149264, 26.612027842415981944958586460360, 27.35437704281223397725874832375, 28.21533447540567617889228326195, 29.39630236879131770500742869289

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