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Properties

Label 2-504-63.5-c1-0-15
Degree $2$
Conductor $504$
Sign $0.917 + 0.398i$
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  + (1.72 − 0.151i)3-s + (−0.793 − 1.37i)5-s + (2.62 + 0.289i)7-s + (2.95 − 0.523i)9-s + (3.42 + 1.97i)11-s + (−5.09 − 2.93i)13-s + (−1.57 − 2.24i)15-s + (3.19 + 5.53i)17-s + (−6.37 − 3.68i)19-s + (4.58 + 0.100i)21-s + (2.10 − 1.21i)23-s + (1.24 − 2.15i)25-s + (5.01 − 1.35i)27-s + (4.34 − 2.50i)29-s − 0.987i·31-s + ⋯
L(s)  = 1  + (0.996 − 0.0875i)3-s + (−0.354 − 0.614i)5-s + (0.993 + 0.109i)7-s + (0.984 − 0.174i)9-s + (1.03 + 0.595i)11-s + (−1.41 − 0.815i)13-s + (−0.407 − 0.580i)15-s + (0.775 + 1.34i)17-s + (−1.46 − 0.844i)19-s + (0.999 + 0.0218i)21-s + (0.438 − 0.253i)23-s + (0.248 − 0.430i)25-s + (0.965 − 0.260i)27-s + (0.806 − 0.465i)29-s − 0.177i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.917 + 0.398i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04352 - 0.424907i\)
\(L(\frac12)\) \(\approx\) \(2.04352 - 0.424907i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 + 0.151i)T \)
7 \( 1 + (-2.62 - 0.289i)T \)
good5 \( 1 + (0.793 + 1.37i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.42 - 1.97i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (5.09 + 2.93i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-3.19 - 5.53i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.37 + 3.68i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.10 + 1.21i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.34 + 2.50i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 0.987iT - 31T^{2} \)
37 \( 1 + (-0.183 + 0.317i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (5.58 - 9.67i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.26 - 2.19i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.08T + 47T^{2} \)
53 \( 1 + (6.89 - 3.97i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 5.46T + 59T^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 + 11.4T + 67T^{2} \)
71 \( 1 + 9.65iT - 71T^{2} \)
73 \( 1 + (-7.64 + 4.41i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + (2.96 + 5.14i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.70 - 8.15i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.4 - 6.05i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.70266436796416162006773646090, −9.883860648665514291614021962336, −8.862040604155155874175821172712, −8.229742327468067800956992704456, −7.54775541336117021233884975732, −6.40191214895010530001928099659, −4.73777531289362250387718374078, −4.30263502290645646933327158253, −2.71880451227106187946603079798, −1.44824469152152340002195720080, 1.74016477567697421728469665002, 3.02303868279659153471457981485, 4.10724554014149707475358611598, 5.06416844784665412312958998719, 6.79447535180832643401543736966, 7.34423143762659723703801069521, 8.326753380302021553280214950704, 9.110811177482382417199659109682, 10.00336535187403776196785669591, 10.96866072158361138022315854680

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