Properties

Label 2-504-63.41-c1-0-9
Degree $2$
Conductor $504$
Sign $0.964 - 0.264i$
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 + (−2.48 − 0.900i)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 + (4.54 − 0.615i)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 + (−0.940 − 0.340i)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 + (0.990 − 0.134i)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

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.964 - 0.264i$
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: \((2,\ 504,\ (\ :1/2),\ 0.964 - 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04473 + 0.140664i\)
\(L(\frac12)\) \(\approx\) \(1.04473 + 0.140664i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.48 + 0.900i)T \)
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

−11.18102920477589671329684450350, −10.06104134886810993076386707769, −9.351585797859589405841341616433, −8.505413780851005541222855515723, −6.73685834013942611799820143853, −6.57962715023284085202196188751, −5.35247194952903431572063094430, −4.19723570395676570348924968400, −3.30288432550840111064164500555, −1.04272726310310475892030590736, 1.01823729344061843353636216209, 2.80622908434257976490638672066, 4.17935932874260592622505510225, 5.51263977935662167831592910373, 6.30141201340277571816060191492, 6.91422871763413486595208983662, 8.193047746972894228242032377805, 9.068881686211388968680082126230, 10.36321511075973510311209900036, 10.69572668041972183567906999452

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