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Properties

Label 2-504-63.47-c1-0-15
Degree $2$
Conductor $504$
Sign $0.993 + 0.115i$
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  + (−1.11 + 1.32i)3-s + 3.53·5-s + (1.30 − 2.30i)7-s + (−0.525 − 2.95i)9-s − 1.81i·11-s + (2.78 − 1.60i)13-s + (−3.93 + 4.69i)15-s + (−2.93 − 5.08i)17-s + (−2.09 − 1.20i)19-s + (1.61 + 4.28i)21-s + 3.84i·23-s + 7.48·25-s + (4.50 + 2.58i)27-s + (5.79 + 3.34i)29-s + (−4.21 − 2.43i)31-s + ⋯
L(s)  = 1  + (−0.642 + 0.766i)3-s + 1.58·5-s + (0.491 − 0.870i)7-s + (−0.175 − 0.984i)9-s − 0.548i·11-s + (0.771 − 0.445i)13-s + (−1.01 + 1.21i)15-s + (−0.711 − 1.23i)17-s + (−0.479 − 0.277i)19-s + (0.351 + 0.936i)21-s + 0.802i·23-s + 1.49·25-s + (0.867 + 0.497i)27-s + (1.07 + 0.620i)29-s + (−0.757 − 0.437i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.993 + 0.115i$
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: \((2,\ 504,\ (\ :1/2),\ 0.993 + 0.115i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.56590 - 0.0906158i\)
\(L(\frac12)\) \(\approx\) \(1.56590 - 0.0906158i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 - 1.32i)T \)
7 \( 1 + (-1.30 + 2.30i)T \)
good5 \( 1 - 3.53T + 5T^{2} \)
11 \( 1 + 1.81iT - 11T^{2} \)
13 \( 1 + (-2.78 + 1.60i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.93 + 5.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.09 + 1.20i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.84iT - 23T^{2} \)
29 \( 1 + (-5.79 - 3.34i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.21 + 2.43i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.905 - 1.56i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-5.03 - 8.71i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.36 - 4.09i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3.96 - 6.86i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.26 + 1.30i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.40 + 7.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (8.98 - 5.18i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.92 + 6.79i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 7.62iT - 71T^{2} \)
73 \( 1 + (11.7 - 6.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.921 - 1.59i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.61 - 9.72i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.89 - 11.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.7 + 7.96i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.01626368406366170269522908629, −10.00521969327591936252492138145, −9.417046500675731985683107021722, −8.461267262072498663737020210020, −6.96577683288074291819640265027, −6.10690369451396628662259002414, −5.29941949589936184591614790414, −4.37966030267338484643330886799, −2.94957080437689505948930002011, −1.16334411624417994819457832611, 1.71433184500570630245320837805, 2.27325711757175545619730776037, 4.50311038264181447474390029296, 5.69431913988287680905106520451, 6.11167877108566351659709690315, 7.00401553264439052822269122494, 8.460054345821891689785785277089, 8.967612554458349828853380144552, 10.33053037019658729158695205675, 10.77352036907634340262015076705

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