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Properties

Label 1-5445-5445.857-r1-0-0
Degree $1$
Conductor $5445$
Sign $-0.901 + 0.432i$
Analytic cond. $585.146$
Root an. cond. $585.146$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.814 + 0.580i)2-s + (0.327 + 0.945i)4-s + (−0.371 − 0.928i)7-s + (−0.281 + 0.959i)8-s + (0.945 − 0.327i)13-s + (0.235 − 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.540 + 0.841i)17-s + (0.841 + 0.540i)19-s + (−0.371 + 0.928i)23-s + (0.959 + 0.281i)26-s + (0.755 − 0.654i)28-s + (−0.0475 + 0.998i)29-s + (0.981 − 0.189i)31-s + (−0.998 + 0.0475i)32-s + ⋯
L(s)  = 1  + (0.814 + 0.580i)2-s + (0.327 + 0.945i)4-s + (−0.371 − 0.928i)7-s + (−0.281 + 0.959i)8-s + (0.945 − 0.327i)13-s + (0.235 − 0.971i)14-s + (−0.786 + 0.618i)16-s + (0.540 + 0.841i)17-s + (0.841 + 0.540i)19-s + (−0.371 + 0.928i)23-s + (0.959 + 0.281i)26-s + (0.755 − 0.654i)28-s + (−0.0475 + 0.998i)29-s + (0.981 − 0.189i)31-s + (−0.998 + 0.0475i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(5445\)    =    \(3^{2} \cdot 5 \cdot 11^{2}\)
Sign: $-0.901 + 0.432i$
Analytic conductor: \(585.146\)
Root analytic conductor: \(585.146\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{5445} (857, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 5445,\ (1:\ ),\ -0.901 + 0.432i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7303675941 + 3.209811030i\)
\(L(\frac12)\) \(\approx\) \(0.7303675941 + 3.209811030i\)
\(L(1)\) \(\approx\) \(1.467853597 + 0.7647861337i\)
\(L(1)\) \(\approx\) \(1.467853597 + 0.7647861337i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (-0.371 - 0.928i)T \)
13 \( 1 + (0.945 - 0.327i)T \)
17 \( 1 + (0.540 + 0.841i)T \)
19 \( 1 + (0.841 + 0.540i)T \)
23 \( 1 + (-0.371 + 0.928i)T \)
29 \( 1 + (-0.0475 + 0.998i)T \)
31 \( 1 + (0.981 - 0.189i)T \)
37 \( 1 + (0.755 - 0.654i)T \)
41 \( 1 + (-0.995 - 0.0950i)T \)
43 \( 1 + (-0.971 + 0.235i)T \)
47 \( 1 + (0.0950 + 0.995i)T \)
53 \( 1 + (-0.989 - 0.142i)T \)
59 \( 1 + (0.580 + 0.814i)T \)
61 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (0.0950 - 0.995i)T \)
71 \( 1 + (-0.841 - 0.540i)T \)
73 \( 1 + (0.989 - 0.142i)T \)
79 \( 1 + (0.723 + 0.690i)T \)
83 \( 1 + (-0.371 - 0.928i)T \)
89 \( 1 + (0.841 - 0.540i)T \)
97 \( 1 + (-0.971 + 0.235i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.630288654222398168878509106857, −16.404861895967671638897352400218, −16.06711295670543672580723764334, −15.3375344360483191759485780621, −14.83269724578670712169927976090, −13.85607213333131395900436032946, −13.57040929767471317540816302697, −12.804871502979270703910814321223, −11.93955364029066777056548459932, −11.711376366121161543826568756242, −11.01443509490602789390286476205, −9.83583288901118958584954463764, −9.80487386690524912114272190871, −8.735091513922347432892891895258, −8.10334016246503528264821186470, −6.83477821038109636170515545103, −6.47169640113148290462077141041, −5.61787325204330058053861514449, −5.07660924580172738565214486672, −4.276012830459190326568510239520, −3.39650069703103694349405172333, −2.81566368769326342927585906448, −2.12980300459620395820235633110, −1.156889910130281546113668786332, −0.344450835298475270832658661997, 0.99881462846246645218113637377, 1.755271015250056144471811376, 3.101863518216933299355686727533, 3.49387216851639320393960954908, 4.10351055761863586542517465338, 5.00657899617500799770591837732, 5.76453222943461634730567754964, 6.33067141241710198084596647641, 7.03488049294722255475189648772, 7.88443996558497317173645329041, 8.14659519938774544574226262568, 9.23138601118046947667966349903, 10.04516406600059669878824172636, 10.75827302682359937808090194401, 11.44690130843071247284567122488, 12.20719907811639803950027593386, 12.89472314453540904778620082732, 13.485827265073513485993148126304, 13.96286517055423583826356420518, 14.656215332230708807157980719062, 15.36498433331765139319234788092, 16.15165580543022015564937771547, 16.40481720017026394299483116709, 17.25404077513439488701856952148, 17.76934457323012622058185485885

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