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Properties

Label 2-1755-65.38-c0-0-4
Degree $2$
Conductor $1755$
Sign $-0.934 + 0.354i$
Analytic cond. $0.875859$
Root an. cond. $0.935873$
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  + (−0.860 − 0.860i)2-s + 0.482i·4-s + (−0.608 + 0.793i)5-s + (−0.445 + 0.445i)8-s + (1.20 − 0.158i)10-s − 0.765i·11-s + (−0.707 − 0.707i)13-s + 1.24·16-s + (−0.382 − 0.293i)20-s + (−0.658 + 0.658i)22-s + (−0.258 − 0.965i)25-s + 1.21i·26-s + (−0.630 − 0.630i)32-s + (−0.0822 − 0.624i)40-s − 1.84i·41-s + ⋯
L(s)  = 1  + (−0.860 − 0.860i)2-s + 0.482i·4-s + (−0.608 + 0.793i)5-s + (−0.445 + 0.445i)8-s + (1.20 − 0.158i)10-s − 0.765i·11-s + (−0.707 − 0.707i)13-s + 1.24·16-s + (−0.382 − 0.293i)20-s + (−0.658 + 0.658i)22-s + (−0.258 − 0.965i)25-s + 1.21i·26-s + (−0.630 − 0.630i)32-s + (−0.0822 − 0.624i)40-s − 1.84i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.354i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $-0.934 + 0.354i$
Analytic conductor: \(0.875859\)
Root analytic conductor: \(0.935873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :0),\ -0.934 + 0.354i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3578763364\)
\(L(\frac12)\) \(\approx\) \(0.3578763364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.608 - 0.793i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
good2 \( 1 + (0.860 + 0.860i)T + iT^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + 0.765iT - T^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + 1.84iT - T^{2} \)
43 \( 1 + (0.366 + 0.366i)T + iT^{2} \)
47 \( 1 + (1.12 + 1.12i)T + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 + 0.261T + T^{2} \)
61 \( 1 + 1.93T + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + 1.58iT - T^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41iT - T^{2} \)
83 \( 1 + (0.184 - 0.184i)T - iT^{2} \)
89 \( 1 - 1.98T + T^{2} \)
97 \( 1 + iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.245364069312182723125410092533, −8.442133441741215045958269987521, −7.82745294930057630974203305898, −6.95774631261504141561984623853, −5.96226419547872830956915414763, −5.07047977482782110608212605559, −3.65629405498977201701786529210, −2.99619240643642883348682505407, −2.01976925310105574504226119519, −0.36857617908029920069103050985, 1.44394414475701743257615342364, 3.04506707919821064627519241789, 4.29916042204123815027719422630, 4.90651317829413650235491083291, 6.12301461126605993012929345027, 6.87671713025332290917164325777, 7.73194643099743322292131097033, 8.027585848684690744496762982684, 9.151321527477391717230542994164, 9.380370907610111917069880181048

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