Properties

Label 2-504-56.27-c1-0-21
Degree $2$
Conductor $504$
Sign $-0.0911 + 0.995i$
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  + (−1.33 + 0.474i)2-s + (1.54 − 1.26i)4-s − 1.58·5-s + (−2.37 + 1.15i)7-s + (−1.46 + 2.42i)8-s + (2.10 − 0.750i)10-s + 2.26·11-s + 0.548·13-s + (2.62 − 2.66i)14-s + (0.798 − 3.91i)16-s − 0.433i·17-s − 6.02i·19-s + (−2.44 + 1.99i)20-s + (−3.01 + 1.07i)22-s − 8.24i·23-s + ⋯
L(s)  = 1  + (−0.941 + 0.335i)2-s + (0.774 − 0.632i)4-s − 0.706·5-s + (−0.899 + 0.436i)7-s + (−0.517 + 0.855i)8-s + (0.665 − 0.237i)10-s + 0.681·11-s + 0.152·13-s + (0.700 − 0.713i)14-s + (0.199 − 0.979i)16-s − 0.105i·17-s − 1.38i·19-s + (−0.547 + 0.447i)20-s + (−0.642 + 0.228i)22-s − 1.71i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0911 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0911 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.0911 + 0.995i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.0911 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.273447 - 0.299622i\)
\(L(\frac12)\) \(\approx\) \(0.273447 - 0.299622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 0.474i)T \)
3 \( 1 \)
7 \( 1 + (2.37 - 1.15i)T \)
good5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 - 2.26T + 11T^{2} \)
13 \( 1 - 0.548T + 13T^{2} \)
17 \( 1 + 0.433iT - 17T^{2} \)
19 \( 1 + 6.02iT - 19T^{2} \)
23 \( 1 + 8.24iT - 23T^{2} \)
29 \( 1 - 0.548iT - 29T^{2} \)
31 \( 1 + 7.50T + 31T^{2} \)
37 \( 1 + 4.21iT - 37T^{2} \)
41 \( 1 + 7.09iT - 41T^{2} \)
43 \( 1 + 1.82T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 3.71iT - 53T^{2} \)
59 \( 1 + 11.5iT - 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 - 9.35T + 67T^{2} \)
71 \( 1 + 1.27iT - 71T^{2} \)
73 \( 1 - 0.867iT - 73T^{2} \)
79 \( 1 + 10.3iT - 79T^{2} \)
83 \( 1 - 6.13iT - 83T^{2} \)
89 \( 1 - 7.95iT - 89T^{2} \)
97 \( 1 - 19.1iT - 97T^{2} \)
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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

−10.64246496881489929644517563395, −9.530326884059829712184153941896, −8.966974677001034224643945975289, −8.114722757194669059043159416513, −6.96220667628299300895983236168, −6.46980692302056316948920992087, −5.20962366243742187966970791392, −3.70058007190576117682845044631, −2.35513972243507150185257457460, −0.33194572095947720914511846655, 1.51110531344219540332982692395, 3.37011598431310275921246860855, 3.89348658680321771848724322685, 5.86185574518273226440246831836, 6.86505648782874494977208272215, 7.66466229643005944254557644416, 8.473289462967331423408819572442, 9.633314654594447027295748861476, 9.981044125946210912265269450092, 11.23279240705269328556536991948

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