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Properties

Label 2-504-63.41-c1-0-19
Degree $2$
Conductor $504$
Sign $0.425 + 0.905i$
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  + (1.53 − 0.803i)3-s + (−0.0977 + 0.169i)5-s + (0.463 − 2.60i)7-s + (1.71 − 2.46i)9-s + (1.54 − 0.890i)11-s + (−5.11 − 2.95i)13-s + (−0.0140 + 0.338i)15-s + 0.588·17-s + 2.48i·19-s + (−1.38 − 4.36i)21-s + (3.85 + 2.22i)23-s + (2.48 + 4.29i)25-s + (0.645 − 5.15i)27-s + (6.28 − 3.62i)29-s + (−3.61 − 2.08i)31-s + ⋯
L(s)  = 1  + (0.886 − 0.463i)3-s + (−0.0437 + 0.0757i)5-s + (0.175 − 0.984i)7-s + (0.570 − 0.821i)9-s + (0.464 − 0.268i)11-s + (−1.41 − 0.819i)13-s + (−0.00362 + 0.0873i)15-s + 0.142·17-s + 0.570i·19-s + (−0.301 − 0.953i)21-s + (0.804 + 0.464i)23-s + (0.496 + 0.859i)25-s + (0.124 − 0.992i)27-s + (1.16 − 0.673i)29-s + (−0.649 − 0.374i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.425 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.425 + 0.905i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (41, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.425 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58823 - 1.00879i\)
\(L(\frac12)\) \(\approx\) \(1.58823 - 1.00879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.53 + 0.803i)T \)
7 \( 1 + (-0.463 + 2.60i)T \)
good5 \( 1 + (0.0977 - 0.169i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.54 + 0.890i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.11 + 2.95i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.588T + 17T^{2} \)
19 \( 1 - 2.48iT - 19T^{2} \)
23 \( 1 + (-3.85 - 2.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.28 + 3.62i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (3.61 + 2.08i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 2.57T + 37T^{2} \)
41 \( 1 + (-0.311 + 0.540i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.08 - 8.80i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.57 - 6.19i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 11.2iT - 53T^{2} \)
59 \( 1 + (5.75 - 9.97i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.57 - 3.79i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.927 + 1.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.58iT - 71T^{2} \)
73 \( 1 + 5.66iT - 73T^{2} \)
79 \( 1 + (-2.92 - 5.06i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.740 - 1.28i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 9.44T + 89T^{2} \)
97 \( 1 + (0.0722 - 0.0417i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.61196170947004882720164561805, −9.808862638891277761093440929281, −8.991030476068778294301598365056, −7.75672387727562520600467465309, −7.46260026616931733273996890004, −6.37568981644495840800141171012, −4.93071475110731730034746599169, −3.75144846388920550495319516260, −2.73801725480509452801164434242, −1.12958839405720355659708161792, 2.06087042672044803642993026846, 2.99123550434340052479334472818, 4.48195968436719028998160467381, 5.10385260136412109430098534572, 6.67612203271892448370229953977, 7.51584669465384736448862264593, 8.808080864069409613663303364944, 9.027056891218665742533740358932, 10.03354280475123888301780150901, 10.94838128492610370274402406600

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