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Properties

Label 2-1452-11.4-c1-0-1
Degree $2$
Conductor $1452$
Sign $-0.859 - 0.511i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 + 0.951i)3-s + (0.809 − 0.587i)5-s + (−0.618 − 1.90i)7-s + (−0.809 − 0.587i)9-s + (−2.42 − 1.76i)13-s + (0.309 + 0.951i)15-s + (−5.66 + 4.11i)17-s + (−1.23 + 3.80i)19-s + 1.99·21-s − 2·23-s + (−1.23 + 3.80i)25-s + (0.809 − 0.587i)27-s + (−0.927 − 2.85i)29-s + (−1.61 − 1.17i)35-s + (2.78 + 8.55i)37-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (0.361 − 0.262i)5-s + (−0.233 − 0.718i)7-s + (−0.269 − 0.195i)9-s + (−0.673 − 0.489i)13-s + (0.0797 + 0.245i)15-s + (−1.37 + 0.997i)17-s + (−0.283 + 0.872i)19-s + 0.436·21-s − 0.417·23-s + (−0.247 + 0.760i)25-s + (0.155 − 0.113i)27-s + (−0.172 − 0.529i)29-s + (−0.273 − 0.198i)35-s + (0.457 + 1.40i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.859 - 0.511i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ -0.859 - 0.511i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5141082383\)
\(L(\frac12)\) \(\approx\) \(0.5141082383\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (-0.809 + 0.587i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.618 + 1.90i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.42 + 1.76i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.66 - 4.11i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (1.23 - 3.80i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + (0.927 + 2.85i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.78 - 8.55i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.78 - 8.55i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + (3.70 - 11.4i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-10.5 - 7.64i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.85 + 5.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.61 - 1.17i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 14T + 67T^{2} \)
71 \( 1 + (-4.85 + 3.52i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (1.85 + 5.70i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (12.9 + 9.40i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.47 - 4.70i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - T + 89T^{2} \)
97 \( 1 + (5.66 + 4.11i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.971429411579498881315617444505, −9.189882089930071780056430659817, −8.282847592626553984494574353800, −7.51226455558187295553921652210, −6.40253629382417084676604779210, −5.82258648283876674398200835246, −4.65994319232331430535098804642, −4.08458013989189222189451266898, −2.93225177104937840293534712109, −1.56999119166723342952710373766, 0.19626116307114708156328443047, 2.16780138146544950222214640498, 2.56476526989992881123280227985, 4.12911715086216059904062673024, 5.15043639332901363434183075320, 5.94347447291200669239165031130, 6.90909171024328893572211765680, 7.22529732881917977518878844090, 8.622700506280142311372622815003, 9.030258546040014983298168952550

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