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Properties

Label 2-504-63.59-c1-0-4
Degree $2$
Conductor $504$
Sign $0.187 - 0.982i$
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 + 0.107i)3-s − 3.60·5-s + (0.415 + 2.61i)7-s + (2.97 + 0.370i)9-s + 4.96i·11-s + (0.167 + 0.0966i)13-s + (−6.23 − 0.387i)15-s + (−0.257 + 0.446i)17-s + (1.69 − 0.979i)19-s + (0.438 + 4.56i)21-s + 5.49i·23-s + 8.01·25-s + (5.10 + 0.960i)27-s + (−6.81 + 3.93i)29-s + (2.78 − 1.60i)31-s + ⋯
L(s)  = 1  + (0.998 + 0.0619i)3-s − 1.61·5-s + (0.157 + 0.987i)7-s + (0.992 + 0.123i)9-s + 1.49i·11-s + (0.0464 + 0.0267i)13-s + (−1.61 − 0.0999i)15-s + (−0.0625 + 0.108i)17-s + (0.389 − 0.224i)19-s + (0.0955 + 0.995i)21-s + 1.14i·23-s + 1.60·25-s + (0.982 + 0.184i)27-s + (−1.26 + 0.730i)29-s + (0.500 − 0.289i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.187 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.187 - 0.982i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.187 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10843 + 0.916902i\)
\(L(\frac12)\) \(\approx\) \(1.10843 + 0.916902i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.72 - 0.107i)T \)
7 \( 1 + (-0.415 - 2.61i)T \)
good5 \( 1 + 3.60T + 5T^{2} \)
11 \( 1 - 4.96iT - 11T^{2} \)
13 \( 1 + (-0.167 - 0.0966i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.257 - 0.446i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.69 + 0.979i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.49iT - 23T^{2} \)
29 \( 1 + (6.81 - 3.93i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.78 + 1.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.09 - 7.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.39 + 5.88i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.07 + 8.78i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-5.78 + 10.0i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.88 + 3.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.80 - 8.32i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.599 - 0.346i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.26 - 2.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8iT - 71T^{2} \)
73 \( 1 + (-5.49 - 3.17i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.99 - 6.91i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.86 + 8.42i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-4.42 - 7.65i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.49 - 1.43i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.30557496479213449814730008820, −10.04036701657104482471113681474, −9.176563971388339688511663175754, −8.405411412762070013040448601341, −7.55318377203252803612956217099, −7.02880378886252098064458888247, −5.20523821097497501797229636771, −4.20644378535383411853306825687, −3.30562083456041110824285767570, −1.98009115900642785141775778730, 0.808698991657039371485115565700, 2.98092317108520289578048373016, 3.81664942456818916678715087595, 4.53147574607219083376252112188, 6.33771091419446176706193414543, 7.52631221882727649836582784594, 7.929893436745482758814605580424, 8.663142378331086882769071213939, 9.755512940799329030008046261493, 10.99095392451973838288765208871

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