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Properties

Label 2-504-168.107-c1-0-2
Degree $2$
Conductor $504$
Sign $-0.0473 - 0.998i$
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  + (−0.648 − 1.25i)2-s + (−1.15 + 1.63i)4-s + (−1.02 + 1.78i)5-s + (1.24 − 2.33i)7-s + (2.80 + 0.396i)8-s + (2.90 + 0.136i)10-s + (−5.24 + 3.03i)11-s − 1.77i·13-s + (−3.74 − 0.0525i)14-s + (−1.31 − 3.77i)16-s + (−0.786 + 0.453i)17-s + (−3.37 + 5.84i)19-s + (−1.71 − 3.73i)20-s + (7.21 + 4.62i)22-s + (−0.351 + 0.608i)23-s + ⋯
L(s)  = 1  + (−0.458 − 0.888i)2-s + (−0.578 + 0.815i)4-s + (−0.459 + 0.796i)5-s + (0.471 − 0.882i)7-s + (0.990 + 0.140i)8-s + (0.918 + 0.0431i)10-s + (−1.58 + 0.913i)11-s − 0.493i·13-s + (−0.999 − 0.0140i)14-s + (−0.329 − 0.944i)16-s + (−0.190 + 0.110i)17-s + (−0.774 + 1.34i)19-s + (−0.383 − 0.835i)20-s + (1.53 + 0.987i)22-s + (−0.0732 + 0.126i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0473 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0473 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0473 - 0.998i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.0473 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282948 + 0.296668i\)
\(L(\frac12)\) \(\approx\) \(0.282948 + 0.296668i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.648 + 1.25i)T \)
3 \( 1 \)
7 \( 1 + (-1.24 + 2.33i)T \)
good5 \( 1 + (1.02 - 1.78i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.24 - 3.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.77iT - 13T^{2} \)
17 \( 1 + (0.786 - 0.453i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.37 - 5.84i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.351 - 0.608i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.52T + 29T^{2} \)
31 \( 1 + (7.31 - 4.22i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-6.07 - 3.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 11.6iT - 41T^{2} \)
43 \( 1 + 4.69T + 43T^{2} \)
47 \( 1 + (-4.42 + 7.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.31 + 5.74i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.11 + 0.645i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.95 - 4.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.562 + 0.974i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 1.43T + 71T^{2} \)
73 \( 1 + (-2.08 - 3.61i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (11.6 + 6.71i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + 4.71iT - 83T^{2} \)
89 \( 1 + (-4.08 - 2.35i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−10.89967309993266850200163902774, −10.42054016727381577208382000296, −9.799967870718632595603963857945, −8.271504563276024345325394822759, −7.74715831340613421849694563168, −7.00982408100220400059864219327, −5.24508946333837386143232619401, −4.14403950260384389179634819974, −3.14952296369838957817583239137, −1.85449512949510158548430683991, 0.27427279536293834125788618119, 2.31792295496842130094614012519, 4.31441726689358666132596906508, 5.23158089707941321286665716670, 5.89619198301890299885498563429, 7.26071005308067628322685223886, 8.119214568874883125330466863372, 8.759044988404912635509301453767, 9.352724731174002699992987780510, 10.79220663205861771288136371953

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