L-function 1-3400-3400.1277-r1-0-0

• Label: 1-3400-3400.1277-r1-0-0
• Degree: $1$
• Conductor: $3400$
• Sign: $-0.648 - 0.760i$
• Analytic cond.: $365.380$
• Root an. cond.: $365.380$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.156 − 0.987i)3^-s + (0.707 − 0.707i)7^-s + (−0.951 + 0.309i)9^-s + (−0.891 + 0.453i)11^-s + (0.951 − 0.309i)13^-s + (−0.587 − 0.809i)19^-s + (−0.809 − 0.587i)21^-s + (−0.453 − 0.891i)23^-s + (0.453 + 0.891i)27^-s + (0.987 − 0.156i)29^-s + (−0.156 + 0.987i)31^-s + (0.587 + 0.809i)33^-s + (0.453 − 0.891i)37^-s + (−0.453 − 0.891i)39^-s + (0.453 − 0.891i)41^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 3400 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.648 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$3400$    =    $2^{3} \cdot 5^{2} \cdot 17$
Sign:	$-0.648 - 0.760i$
Analytic conductor:	$365.380$
Root analytic conductor:	$365.380$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{3400} (1277, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 3400,\ (1:\ ),\ -0.648 - 0.760i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.9173092896 - 1.988245359i$
$L(1)$	$\approx$	$0.9632330396 - 0.5047186165i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	5	$1$
	17	$1$
good	3	$1 + (-0.156 - 0.987i)T$
	7	$1 + (0.707 - 0.707i)T$
	11	$1 + (-0.891 + 0.453i)T$
	13	$1 + (0.951 - 0.309i)T$
	19	$1 + (-0.587 - 0.809i)T$
	23	$1 + (-0.453 - 0.891i)T$
	29	$1 + (0.987 - 0.156i)T$
	31	$1 + (-0.156 + 0.987i)T$
	37	$1 + (0.453 - 0.891i)T$

Imaginary part of the first few zeros on the critical line

−18.82579161232560488058442024141, −18.05834951385777826687800230901, −17.64057885134272345016651690632, −16.488308591496242392642214718687, −16.25626705081818186396786505073, −15.34069886073089228159406396304, −14.98980536232705935487400406261, −14.11377830917768812991582523671, −13.47369297897690882315025030893, −12.543243232855425922582373683180, −11.54296305104090101594315204693, −11.3299813745934167836248651438, −10.438307249726417629632136977397, −9.84625524135173418266799279213, −8.94527650215857565818957338286, −8.30836358887224861779954888364, −7.87122316406568858945352368535, −6.435592006227046393437803224974, −5.79298260604923946585546909343, −5.252948414278944528033447712534, −4.37745335213254065396205085286, −3.69354681294614431999994507064, −2.79236907827227045029281090367, −1.9677758708129226753929220967, −0.79649605895709794514826697877, 0.470232100345537002598883595546, 1.02905788891700511737135349244, 2.1082591767972318576143642864, 2.660697381980247607303828552414, 3.85140341875251793873374798786, 4.70282816776292030128467740219, 5.44767911864896240219312326710, 6.31264426283455351391555906870, 7.011483875936993268959033349639, 7.68926687837881708425674486439, 8.325530224088494070882610184602, 8.89826859987928880704078294833, 10.228270725250579740967918731180, 10.792419831700105161041889272579, 11.270789723730008016351963338687, 12.2895471466796596670431792512, 12.830020391362235650000441801901, 13.46383793740652931345687973322, 14.14580767977975884589192291544, 14.70431034157827550578784992988, 15.7771158828432341622144973161, 16.28019912587925630821950706955, 17.37011931262223276212133169798, 17.74856652732114360148166547503, 18.18035357850165333069165177027

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