L-function 1-319-319.10-r0-0-0

• Label: 1-319-319.10-r0-0-0
• Degree: $1$
• Conductor: $319$
• Sign: $-0.853 + 0.521i$
• Analytic cond.: $1.48142$
• Root an. cond.: $1.48142$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (−0.433 + 0.900i)2^-s + (0.781 + 0.623i)3^-s + (−0.623 − 0.781i)4^-s + (0.900 + 0.433i)5^-s + (−0.900 + 0.433i)6^-s + (−0.623 + 0.781i)7^-s + (0.974 − 0.222i)8^-s + (0.222 + 0.974i)9^-s + (−0.781 + 0.623i)10^-s − i·12^-s + (−0.222 + 0.974i)13^-s + (−0.433 − 0.900i)14^-s + (0.433 + 0.900i)15^-s + (−0.222 + 0.974i)16^-s − i·17^-s + (−0.974 − 0.222i)18^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 319 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.521i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$319$    =    $11 \cdot 29$
Sign:	$-0.853 + 0.521i$
Analytic conductor:	$1.48142$
Root analytic conductor:	$1.48142$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{319} (10, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 319,\ (0:\ ),\ -0.853 + 0.521i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$0.3529600715 + 1.254131298i$
$L(1)$	$\approx$	$0.7692763753 + 0.8086545298i$

Euler product

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

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

Imaginary part of the first few zeros on the critical line

−25.04207476684340973871779897969, −24.01475395827071206832266561569, −22.82716109078162802254014657507, −21.935007089094350137295538214361, −20.802054035490924427200092847637, −20.28859216940405294054051864954, −19.57473379965921716482989792735, −18.63939576236626102479692013628, −17.66253180400903467270133205491, −17.11633945705958064351879292365, −15.853536714229150745929479111868, −14.176461701551902832922823798289, −13.7041491049133549157893420402, −12.67032047500674900627341518056, −12.31894688301142967820556441787, −10.43737490605532418337187459835, −10.0196390172163114971438432934, −8.93569821182328143486757365056, −8.09559913128448535524383034706, −7.05571757910898308806194193390, −5.69612155603500112971830026459, −4.04785699933730836650096602429, −3.09845580277616490084140226417, −1.963857656664691627547443525903, −0.91685113363123022106528840300, 1.93348262372945554630436639440, 3.03604170773066841505000125416, 4.59573445601925365626367851842, 5.60324446675486875793711147085, 6.62644778824183304287443333954, 7.64368358772397755372944612526, 8.97969502531370941613491740565, 9.48205996684280012863518646603, 10.080846092734776999482845598447, 11.48964365474974847574228561747, 13.24031110212641889287846173733, 13.937241809949079503555772054564, 14.63653203053287509958848681358, 15.693690496471133315850321249958, 16.207855951133635529142877817418, 17.33931663717584540182404587076, 18.40424961717881004821348588720, 19.00673623457785535182325402303, 19.99795976695568058933379720654, 21.21354485503807062268754140469, 22.122155339628694866958936078334, 22.58549195246271335400451128270, 24.17253949403403499344882479637, 24.89854729645083837768877228334, 25.64892570571564758004025143629

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