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Properties

Label 2-504-8.5-c1-0-9
Degree $2$
Conductor $504$
Sign $0.947 + 0.321i$
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  + (−0.569 − 1.29i)2-s + (−1.35 + 1.47i)4-s + 0.772i·5-s − 7-s + (2.67 + 0.908i)8-s + (1 − 0.440i)10-s − 4.40i·11-s + 5.89i·13-s + (0.569 + 1.29i)14-s + (−0.350 − 3.98i)16-s + 4.21·17-s + 5.01i·19-s + (−1.13 − 1.04i)20-s + (−5.70 + 2.50i)22-s + 8.77·23-s + ⋯
L(s)  = 1  + (−0.402 − 0.915i)2-s + (−0.675 + 0.737i)4-s + 0.345i·5-s − 0.377·7-s + (0.947 + 0.321i)8-s + (0.316 − 0.139i)10-s − 1.32i·11-s + 1.63i·13-s + (0.152 + 0.345i)14-s + (−0.0876 − 0.996i)16-s + 1.02·17-s + 1.15i·19-s + (−0.254 − 0.233i)20-s + (−1.21 + 0.535i)22-s + 1.82·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.321i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.321i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.947 + 0.321i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (253, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.947 + 0.321i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05483 - 0.173934i\)
\(L(\frac12)\) \(\approx\) \(1.05483 - 0.173934i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.569 + 1.29i)T \)
3 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 0.772iT - 5T^{2} \)
11 \( 1 + 4.40iT - 11T^{2} \)
13 \( 1 - 5.89iT - 13T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
19 \( 1 - 5.01iT - 19T^{2} \)
23 \( 1 - 8.77T + 23T^{2} \)
29 \( 1 - 3.63iT - 29T^{2} \)
31 \( 1 - 7.40T + 31T^{2} \)
37 \( 1 + 5.01iT - 37T^{2} \)
41 \( 1 + 8.77T + 41T^{2} \)
43 \( 1 - 0.880iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6.72iT - 53T^{2} \)
59 \( 1 + 10.3iT - 59T^{2} \)
61 \( 1 - 5.89iT - 61T^{2} \)
67 \( 1 - 0.880iT - 67T^{2} \)
71 \( 1 + 4.21T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + 1.54iT - 83T^{2} \)
89 \( 1 - 8.77T + 89T^{2} \)
97 \( 1 - 16.8T + 97T^{2} \)
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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

−10.86706312864014580614465125928, −10.11949668587622685930120706975, −9.111492046378453251985740606176, −8.542119937164810181549597837463, −7.35998393445539924049029264592, −6.35464204147607771092255943731, −5.00720872725368538376112579821, −3.68240926516034957401925997177, −2.91048722861665329930037297635, −1.26031584173127557723998357608, 0.922568715449747197900600226496, 3.01218351171080009070782051668, 4.73332308725475268925012887396, 5.27015362040381311688504260823, 6.56276231112118442490375284614, 7.32443710734900379526653080308, 8.202800753413010619257095255254, 9.101387032191566374167970290753, 9.999190142673661559548412441683, 10.51507952584987395518939186356

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