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Properties

Label 2-504-168.11-c1-0-22
Degree $2$
Conductor $504$
Sign $-0.955 + 0.294i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (−0.830 − 1.14i)2-s + (−0.620 + 1.90i)4-s + (−1.88 − 3.26i)5-s + (2.23 + 1.41i)7-s + (2.69 − 0.868i)8-s + (−2.16 + 4.86i)10-s + (1.47 + 0.849i)11-s − 5.64i·13-s + (−0.233 − 3.73i)14-s + (−3.23 − 2.35i)16-s + (−2.26 − 1.30i)17-s + (−1.18 − 2.04i)19-s + (7.36 − 1.55i)20-s + (−0.249 − 2.38i)22-s + (0.653 + 1.13i)23-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.310 + 0.950i)4-s + (−0.841 − 1.45i)5-s + (0.844 + 0.535i)7-s + (0.951 − 0.307i)8-s + (−0.685 + 1.53i)10-s + (0.443 + 0.256i)11-s − 1.56i·13-s + (−0.0624 − 0.998i)14-s + (−0.807 − 0.589i)16-s + (−0.548 − 0.316i)17-s + (−0.270 − 0.469i)19-s + (1.64 − 0.347i)20-s + (−0.0532 − 0.509i)22-s + (0.136 + 0.236i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.955 + 0.294i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.955 + 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.108847 - 0.722281i\)
\(L(\frac12)\) \(\approx\) \(0.108847 - 0.722281i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.830 + 1.14i)T \)
3 \( 1 \)
7 \( 1 + (-2.23 - 1.41i)T \)
good5 \( 1 + (1.88 + 3.26i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.47 - 0.849i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.64iT - 13T^{2} \)
17 \( 1 + (2.26 + 1.30i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.18 + 2.04i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.653 - 1.13i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.80T + 29T^{2} \)
31 \( 1 + (4.75 + 2.74i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.20 + 3.00i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.10iT - 41T^{2} \)
43 \( 1 + 6.49T + 43T^{2} \)
47 \( 1 + (3.53 + 6.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.488 + 0.846i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.67 - 3.85i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.64 - 4.41i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.69 + 11.6i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 9.40T + 71T^{2} \)
73 \( 1 + (1.09 - 1.88i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-15.0 + 8.71i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.8iT - 83T^{2} \)
89 \( 1 + (-9.58 + 5.53i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.0T + 97T^{2} \)
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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

−10.69356537941261533353742156209, −9.373148497323278413584647709307, −8.837244034368664359646981570358, −8.056045116040542442821683414551, −7.43013706554876488900453673888, −5.43388038930525167905923660932, −4.64009474967710528871236241769, −3.60694186126422026556463159430, −1.97431326363350028931055110170, −0.54268674564894637676010221459, 1.83650810470303372112639749009, 3.76012503582615340744849187227, 4.62648284238103509568540394836, 6.24513755497818433540765773930, 6.87982779253982974050227317652, 7.59813578012541095914929887352, 8.425110100452585092889256087807, 9.453135723214534336903545425759, 10.52008007663499299024478729597, 11.22692362890624102621049448877

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