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Properties

Label 2-504-63.47-c1-0-0
Degree $2$
Conductor $504$
Sign $-0.710 + 0.703i$
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  + (0.149 + 1.72i)3-s − 2.22·5-s + (−2.45 + 0.996i)7-s + (−2.95 + 0.514i)9-s − 1.17i·11-s + (3.12 − 1.80i)13-s + (−0.331 − 3.83i)15-s + (−3.71 − 6.42i)17-s + (−3.05 − 1.76i)19-s + (−2.08 − 4.08i)21-s + 5.81i·23-s − 0.0674·25-s + (−1.32 − 5.02i)27-s + (−6.04 − 3.48i)29-s + (6.88 + 3.97i)31-s + ⋯
L(s)  = 1  + (0.0860 + 0.996i)3-s − 0.993·5-s + (−0.926 + 0.376i)7-s + (−0.985 + 0.171i)9-s − 0.353i·11-s + (0.866 − 0.500i)13-s + (−0.0854 − 0.989i)15-s + (−0.900 − 1.55i)17-s + (−0.700 − 0.404i)19-s + (−0.454 − 0.890i)21-s + 1.21i·23-s − 0.0134·25-s + (−0.255 − 0.966i)27-s + (−1.12 − 0.648i)29-s + (1.23 + 0.713i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.710 + 0.703i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (425, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.710 + 0.703i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00838870 - 0.0203840i\)
\(L(\frac12)\) \(\approx\) \(0.00838870 - 0.0203840i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.149 - 1.72i)T \)
7 \( 1 + (2.45 - 0.996i)T \)
good5 \( 1 + 2.22T + 5T^{2} \)
11 \( 1 + 1.17iT - 11T^{2} \)
13 \( 1 + (-3.12 + 1.80i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.71 + 6.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.05 + 1.76i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.81iT - 23T^{2} \)
29 \( 1 + (6.04 + 3.48i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-6.88 - 3.97i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.54 - 9.60i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.809 + 1.40i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.904 + 1.56i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.26 + 7.38i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (9.62 - 5.55i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.00 - 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (7.09 - 4.09i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.96 - 8.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.67iT - 71T^{2} \)
73 \( 1 + (-6.92 + 3.99i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.25 - 3.91i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.390 - 0.677i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.75 + 3.03i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.49 + 2.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.47600618181816729882200963558, −10.61737233009624288369716138552, −9.628702004165332838312070859429, −8.891562421584014686504867693928, −8.119055137079458118485806738907, −6.88655191204807930653643587552, −5.81334785877350408687562280322, −4.71594780450404335752645821429, −3.64218799802650222335829513785, −2.89855737432440314504560880824, 0.01237956468852625754538608839, 1.89973590185831660008776093807, 3.47682157873533439074917060608, 4.29500722781152556929003897235, 6.20841817181249875844709050897, 6.54989715288386481859439890743, 7.69291972367896683764282648087, 8.384381346889111065903804217704, 9.257353092506119807532495070404, 10.67443810484560544119496561619

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