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Properties

Label 2-504-56.53-c1-0-27
Degree $2$
Conductor $504$
Sign $0.392 + 0.919i$
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  + (0.242 + 1.39i)2-s + (−1.88 + 0.674i)4-s + (−1.28 + 0.742i)5-s + (0.129 − 2.64i)7-s + (−1.39 − 2.45i)8-s + (−1.34 − 1.61i)10-s + (−4.37 − 2.52i)11-s − 2.58i·13-s + (3.71 − 0.459i)14-s + (3.08 − 2.54i)16-s + (0.629 − 1.09i)17-s + (−2.68 + 1.54i)19-s + (1.92 − 2.26i)20-s + (2.45 − 6.70i)22-s + (−0.697 − 1.20i)23-s + ⋯
L(s)  = 1  + (0.171 + 0.985i)2-s + (−0.941 + 0.337i)4-s + (−0.575 + 0.332i)5-s + (0.0490 − 0.998i)7-s + (−0.493 − 0.869i)8-s + (−0.425 − 0.510i)10-s + (−1.31 − 0.760i)11-s − 0.717i·13-s + (0.992 − 0.122i)14-s + (0.772 − 0.635i)16-s + (0.152 − 0.264i)17-s + (−0.615 + 0.355i)19-s + (0.429 − 0.506i)20-s + (0.523 − 1.42i)22-s + (−0.145 − 0.252i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.392 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.392 + 0.919i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.392 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.453869 - 0.299663i\)
\(L(\frac12)\) \(\approx\) \(0.453869 - 0.299663i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.242 - 1.39i)T \)
3 \( 1 \)
7 \( 1 + (-0.129 + 2.64i)T \)
good5 \( 1 + (1.28 - 0.742i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.37 + 2.52i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.58iT - 13T^{2} \)
17 \( 1 + (-0.629 + 1.09i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.68 - 1.54i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.697 + 1.20i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.638iT - 29T^{2} \)
31 \( 1 + (-1.82 + 3.16i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.21 + 3.01i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 + 1.02iT - 43T^{2} \)
47 \( 1 + (5.48 + 9.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.99 - 2.88i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.01 + 1.74i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (11.1 - 6.44i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.443 + 0.256i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.41T + 71T^{2} \)
73 \( 1 + (4.94 - 8.56i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.35 + 7.54i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2.97iT - 83T^{2} \)
89 \( 1 + (1.29 + 2.23i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 1.57T + 97T^{2} \)
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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

−10.58798072111657015255619889259, −9.966879902495547402318833980335, −8.553778746987698343716993908623, −7.84593622845477678875258665881, −7.30571812865965325260384201884, −6.14954647329178335164493716893, −5.19341577510378354659702262405, −4.06880471896005009901298647873, −3.10367825442088033274597227906, −0.29923478454618090034110518133, 1.92339529370894945842377220208, 2.97920384809167146283671589332, 4.42033354191570083319748493744, 5.06182315909894055802622211023, 6.26046060048401613766958228405, 7.83103337263539829058142192088, 8.521773808857688932172960885003, 9.444922093678807890468198198966, 10.27046400516464927790203796384, 11.21954541640000215083119653231

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