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Properties

Label 2-504-63.38-c1-0-3
Degree $2$
Conductor $504$
Sign $-0.435 - 0.900i$
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  + (−0.445 − 1.67i)3-s + (−0.101 + 0.175i)5-s + (−1.37 + 2.26i)7-s + (−2.60 + 1.49i)9-s + (−3.86 + 2.23i)11-s + (−1.25 + 0.725i)13-s + (0.339 + 0.0916i)15-s + (1.60 − 2.78i)17-s + (−6.20 + 3.58i)19-s + (4.39 + 1.29i)21-s + (−1.09 − 0.633i)23-s + (2.47 + 4.29i)25-s + (3.65 + 3.69i)27-s + (−0.944 − 0.545i)29-s − 6.46i·31-s + ⋯
L(s)  = 1  + (−0.257 − 0.966i)3-s + (−0.0454 + 0.0786i)5-s + (−0.519 + 0.854i)7-s + (−0.867 + 0.497i)9-s + (−1.16 + 0.673i)11-s + (−0.348 + 0.201i)13-s + (0.0877 + 0.0236i)15-s + (0.389 − 0.674i)17-s + (−1.42 + 0.821i)19-s + (0.959 + 0.281i)21-s + (−0.228 − 0.132i)23-s + (0.495 + 0.858i)25-s + (0.703 + 0.710i)27-s + (−0.175 − 0.101i)29-s − 1.16i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.435 - 0.900i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.435 - 0.900i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.202257 + 0.322386i\)
\(L(\frac12)\) \(\approx\) \(0.202257 + 0.322386i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.445 + 1.67i)T \)
7 \( 1 + (1.37 - 2.26i)T \)
good5 \( 1 + (0.101 - 0.175i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.86 - 2.23i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.25 - 0.725i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-1.60 + 2.78i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.20 - 3.58i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.09 + 0.633i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.944 + 0.545i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 6.46iT - 31T^{2} \)
37 \( 1 + (-3.02 - 5.24i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.370 - 0.642i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.69 - 8.13i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.0931T + 47T^{2} \)
53 \( 1 + (9.35 + 5.39i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.3T + 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 8.05T + 67T^{2} \)
71 \( 1 + 15.6iT - 71T^{2} \)
73 \( 1 + (-0.984 - 0.568i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 11.7T + 79T^{2} \)
83 \( 1 + (-2.29 + 3.98i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.52 - 6.10i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.17 - 1.83i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.41215372083751579797826024877, −10.32621014591431982529136973319, −9.450722919378414662719760819051, −8.296902689166713670137695769781, −7.61788509432911155087748286547, −6.59232575399943279055955316401, −5.77141244209015685966657071004, −4.78223187120693837543870076097, −2.95070505936677887433761058917, −2.01451508415341980655562070943, 0.21644197818458712178004532831, 2.78318535569092666144853435832, 3.86665451019359194973627967292, 4.84192913015754014289919791034, 5.85368096280213218313660877721, 6.86194069131336897952293032917, 8.122539792549103992473697601842, 8.871567712121886746228979985413, 10.07159253582879736893479763953, 10.53651936236590288766764608835

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