L-function 1-624-624.5-r0-0-0

• Label: 1-624-624.5-r0-0-0
• Degree: $1$
• Conductor: $624$
• Sign: $-0.0985 - 0.995i$
• Analytic cond.: $2.89784$
• Root an. cond.: $2.89784$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

− 5^-s − i·7^-s + 11^-s + 17^-s − 19^-s − 23^-s + 25^-s + i·29^-s − i·31^-s + i·35^-s − 37^-s − i·41^-s − i·43^-s − i·47^-s − 49^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0985 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(624\)    =    \(2^{4} \cdot 3 \cdot 13\)
Sign:	$-0.0985 - 0.995i$
Analytic conductor:	\(2.89784\)
Root analytic conductor:	\(2.89784\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{624} (5, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 624,\ (0:\ ),\ -0.0985 - 0.995i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.6261309185 - 0.6911921743i\)
\(L(1)\)	\(\approx\)	\(0.8509959789 - 0.2157075238i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	3	\( 1 \)
	13	\( 1 \)
good	5	\( 1 \)
	7	\( 1 \)
	11	\( 1 \)
	17	\( 1 \)
	19	\( 1 - iT \)
	23	\( 1 \)
	29	\( 1 \)
	31	\( 1 \)
	37	\( 1 + T \)

Imaginary part of the first few zeros on the critical line

−23.14481400861195782792835444098, −22.45051581409248478608891745742, −21.58880378631513530253650065631, −20.79404775289003631517237177970, −19.61018483654668884103311089495, −19.25842692443185673527844000224, −18.39596809669110040185599132240, −17.395436498779143199467671984774, −16.416653593321604315225686199671, −15.721807420787136431101976119196, −14.83178755088993439171069641911, −14.28294606179183609188401256453, −12.873099042829045352441902391337, −12.02895978561225839475538220364, −11.67337652855101339607938110981, −10.50218002521158872692114939227, −9.41255163104746797636581193442, −8.52475815587243266813663555132, −7.84529965489808921211647406848, −6.66085101036855433878056380298, −5.82542763057087788774118758244, −4.60504658724480661726510071731, −3.73844577272434005685754615340, −2.69491203219799001828366640842, −1.350381854804742612684132572767, 0.506868806094858540992572105118, 1.83551888577640330267251983637, 3.6000830760014325583298848305, 3.88137173594213435856578680159, 5.04789698354245653799948453158, 6.42051053698386636234961143454, 7.2090915971965090773495228290, 8.03229053732915612995861886526, 8.93170458690156977512390595918, 10.13098927104985003790701175787, 10.84810184856667750779566422763, 11.85542044851489462660973881556, 12.444705183195892843882518268691, 13.634986929132663654653932259954, 14.45210538298930180129964276706, 15.15854545415101705060480440651, 16.32256780848056053368715932267, 16.77578752836875979100028398289, 17.69370192234626958185386391006, 18.91398777309642573128344523940, 19.45460901679798849426175842000, 20.23691633119123602655389465674, 20.92217828644445291709250820791, 22.17889540618814404041647234100, 22.762862277553871440148269951

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