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Properties

Label 2-507-507.449-c1-0-55
Degree $2$
Conductor $507$
Sign $-0.652 + 0.757i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (0.481 − 1.66i)3-s + (0.160 − 1.99i)4-s + (1.32 − 3.28i)7-s + (−2.53 − 1.60i)9-s + (−3.23 − 1.22i)12-s + (2.59 + 2.5i)13-s + (−3.94 − 0.641i)16-s + (−1.71 + 6.38i)19-s + (−4.83 − 3.78i)21-s + (3.31 − 3.74i)25-s + (−3.88 + 3.44i)27-s + (−6.34 − 3.16i)28-s + (8.43 + 0.509i)31-s + (−3.60 + 4.79i)36-s + (−0.595 − 0.393i)37-s + ⋯
L(s)  = 1  + (0.278 − 0.960i)3-s + (0.0804 − 0.996i)4-s + (0.500 − 1.24i)7-s + (−0.845 − 0.534i)9-s + (−0.935 − 0.354i)12-s + (0.720 + 0.693i)13-s + (−0.987 − 0.160i)16-s + (−0.392 + 1.46i)19-s + (−1.05 − 0.826i)21-s + (0.663 − 0.748i)25-s + (−0.748 + 0.663i)27-s + (−1.19 − 0.598i)28-s + (1.51 + 0.0915i)31-s + (−0.600 + 0.799i)36-s + (−0.0978 − 0.0646i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.652 + 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.652 + 0.757i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.652 + 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.655250 - 1.42968i\)
\(L(\frac12)\) \(\approx\) \(0.655250 - 1.42968i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.481 + 1.66i)T \)
13 \( 1 + (-2.59 - 2.5i)T \)
good2 \( 1 + (-0.160 + 1.99i)T^{2} \)
5 \( 1 + (-3.31 + 3.74i)T^{2} \)
7 \( 1 + (-1.32 + 3.28i)T + (-5.04 - 4.84i)T^{2} \)
11 \( 1 + (9.93 + 4.71i)T^{2} \)
17 \( 1 + (11.7 - 12.2i)T^{2} \)
19 \( 1 + (1.71 - 6.38i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (28.9 + 2.33i)T^{2} \)
31 \( 1 + (-8.43 - 0.509i)T + (30.7 + 3.73i)T^{2} \)
37 \( 1 + (0.595 + 0.393i)T + (14.5 + 34.0i)T^{2} \)
41 \( 1 + (-21.9 + 34.6i)T^{2} \)
43 \( 1 + (12.4 + 2.53i)T + (39.5 + 16.8i)T^{2} \)
47 \( 1 + (-43.9 - 16.6i)T^{2} \)
53 \( 1 + (51.4 + 12.6i)T^{2} \)
59 \( 1 + (18.6 + 55.9i)T^{2} \)
61 \( 1 + (-12.4 + 9.38i)T + (16.9 - 58.5i)T^{2} \)
67 \( 1 + (9.72 + 8.27i)T + (10.7 + 66.1i)T^{2} \)
71 \( 1 + (70.9 + 2.85i)T^{2} \)
73 \( 1 + (-15.4 + 4.82i)T + (60.0 - 41.4i)T^{2} \)
79 \( 1 + (-8.45 - 12.2i)T + (-28.0 + 73.8i)T^{2} \)
83 \( 1 + (38.5 - 73.4i)T^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-9.88 + 0.998i)T + (95.0 - 19.4i)T^{2} \)
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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

−10.60086618929985394703590286355, −9.857793504967171779169924160404, −8.610173684348972830924271923064, −7.899051839017749073526252554147, −6.71808903598406892054690505125, −6.30042914378113535551964656471, −4.93267149933173287034433190533, −3.73294893211674427748212725106, −1.96049179930510177908478812543, −0.969850616718179654932326785575, 2.49250542377729939206399310585, 3.27563918707544424642827960233, 4.55529842616851712604650764097, 5.38265162782170473112181741571, 6.65361236373256048675570735614, 8.045180096033822950670401775156, 8.605067572369994087447214277949, 9.175804973894991833052284549558, 10.39616784268130336880827205024, 11.38110237254893621273926181268

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