L-function 1-2652-2652.887-r1-0-0

• Label: 1-2652-2652.887-r1-0-0
• Degree: $1$
• Conductor: $2652$
• Sign: $-0.552 + 0.833i$
• Analytic cond.: $284.996$
• Root an. cond.: $284.996$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ (0.382 − 0.923i)5^-s + (0.608 − 0.793i)7^-s + (0.130 − 0.991i)11^-s + (0.965 + 0.258i)19^-s + (−0.130 + 0.991i)23^-s + (−0.707 − 0.707i)25^-s + (−0.608 − 0.793i)29^-s + (0.923 + 0.382i)31^-s + (−0.5 − 0.866i)35^-s + (−0.793 + 0.608i)37^-s + (−0.991 − 0.130i)41^-s + (−0.258 + 0.965i)43^-s − i·47^-s + (−0.258 − 0.965i)49^-s + (−0.707 + 0.707i)53^-s + ⋯

Functional equation

$\begin{aligned}\Lambda(s)=\mathstrut & 2652 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.552 + 0.833i)\, \overline{\Lambda}(1-s) \end{aligned}$

Invariants

Degree:	$1$
Conductor:	$2652$    =    $2^{2} \cdot 3 \cdot 13 \cdot 17$
Sign:	$-0.552 + 0.833i$
Analytic conductor:	$284.996$
Root analytic conductor:	$284.996$
Motivic weight:	$0$
Rational:	no
Arithmetic:	yes
Character:	$\chi_{2652} (887, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	$0$
Selberg data:	$(1,\ 2652,\ (1:\ ),\ -0.552 + 0.833i)$

Particular Values

$L(\frac{1}{2})$	$\approx$	$-0.06749890898 - 0.1257215021i$
$L(1)$	$\approx$	$1.019624478 - 0.3208174236i$

Euler product

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

	$p$	$F_p(T)$
bad	2	$1$
	3	$1$
	13	$1$
	17	$1$
good	5	$1 + (0.382 - 0.923i)T$
	7	$1 + (0.608 - 0.793i)T$
	11	$1 + (0.130 - 0.991i)T$
	19	$1 + (0.965 + 0.258i)T$
	23	$1 + (-0.130 + 0.991i)T$
	29	$1 + (-0.608 - 0.793i)T$
	31	$1 + (0.923 + 0.382i)T$
	37	$1 + (-0.793 + 0.608i)T$
	41	$1 + (-0.991 - 0.130i)T$

Imaginary part of the first few zeros on the critical line

−19.538400666400233974168769735016, −18.66727257557683212890490267638, −18.24812890107866126813846536609, −17.65653560845930217957794099576, −16.97840921427742046041222452924, −15.89846784325335273437151804836, −15.22512081585649483407992160729, −14.69810728349784531577491075047, −14.084258093614271759845302315218, −13.28695414215505750790571966357, −12.290490981611573546673162093, −11.79824352674957445357632499063, −10.981638940274411894889259981785, −10.23396084202189667995628229347, −9.572970773523854588867426706486, −8.77713497340459432521054367984, −7.91780239374519707077293171637, −7.06310299307821733241496120984, −6.53966000585706664376794515410, −5.47912912159124081090922671227, −4.978552860608855066834405809552, −3.88250841398502188837128285249, −2.93237821972940931619783790792, −2.17608410588366243346952101095, −1.51747352754382648648584110099, 0.02132765666336723807259592743, 1.18471022775783140802013263547, 1.46758179964894223265686522250, 2.873896216989717586948307504197, 3.76447645199973913666675628127, 4.5693748509983867588752228046, 5.33956359307244831079102709227, 5.97509759509829754134291756908, 6.97824574707686378679981285861, 7.96484886194575321882776157750, 8.312313873147049571946181187340, 9.34352860973675710806568620767, 9.89113997310923563009006078264, 10.80360593732178219806418737852, 11.5795232310524162161181529484, 12.11531155635881201379714502828, 13.222948911578890749503478950981, 13.71400962598537740519011788716, 14.112585963690380406008437380910, 15.21409899211111797010574835889, 16.033759098767343645597348139728, 16.59182376303766583224571881852, 17.33223669602695794695128276037, 17.71234688860321519777155780920, 18.73964294869320440721693961890

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