Properties

Degree $2$
Conductor $504$
Sign $0.386 - 0.922i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1 + 1.73i)5-s + (2.5 + 0.866i)7-s + (−3 + 5.19i)11-s − 3·13-s + (2 − 3.46i)17-s + (2.5 + 4.33i)19-s + (−2 − 3.46i)23-s + (0.500 − 0.866i)25-s + 4·29-s + (−3.5 + 6.06i)31-s + (1.00 + 5.19i)35-s + (4.5 + 7.79i)37-s + 2·41-s − 43-s + (1 + 1.73i)47-s + ⋯
L(s)  = 1  + (0.447 + 0.774i)5-s + (0.944 + 0.327i)7-s + (−0.904 + 1.56i)11-s − 0.832·13-s + (0.485 − 0.840i)17-s + (0.573 + 0.993i)19-s + (−0.417 − 0.722i)23-s + (0.100 − 0.173i)25-s + 0.742·29-s + (−0.628 + 1.08i)31-s + (0.169 + 0.878i)35-s + (0.739 + 1.28i)37-s + 0.312·41-s − 0.152·43-s + (0.145 + 0.252i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $0.386 - 0.922i$
Motivic weight: \(1\)
Character: $\chi_{504} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ 0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27680 + 0.849305i\)
\(L(\frac12)\) \(\approx\) \(1.27680 + 0.849305i\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (-2.5 - 0.866i)T \)
good5 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (3 - 5.19i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3T + 13T^{2} \)
17 \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + (3.5 - 6.06i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.5 - 7.79i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 + T + 43T^{2} \)
47 \( 1 + (-1 - 1.73i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4 + 6.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5 + 8.66i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 12.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (4 + 6.92i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−10.94494209126543165424406328632, −10.10634625282412151359127238594, −9.645974251661760253846192525462, −8.178880345957336238189269793949, −7.50970555769538460703916113559, −6.61226052008171189603554239903, −5.27210085663573538402497997387, −4.66164764227298881821069566165, −2.87446724735790936959966011728, −1.94480432129522434333023636780, 0.972834040942710151842585743168, 2.55391115072584950497414244288, 4.06104830099305256895123266484, 5.30595750339914328143036925768, 5.70474726903762060798757274996, 7.34413989000089131159784952071, 8.098329698175852775873073983742, 8.890528885536642797470228626488, 9.852024814198108807289053012031, 10.87072075275514347279479372708

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