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Properties

Label 2-506-11.5-c1-0-16
Degree $2$
Conductor $506$
Sign $0.360 - 0.932i$
Analytic cond. $4.04043$
Root an. cond. $2.01008$
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  + (0.309 + 0.951i)2-s + (2.27 + 1.65i)3-s + (−0.809 + 0.587i)4-s + (0.695 − 2.14i)5-s + (−0.868 + 2.67i)6-s + (2.50 − 1.82i)7-s + (−0.809 − 0.587i)8-s + (1.51 + 4.65i)9-s + 2.25·10-s + (1.92 + 2.70i)11-s − 2.81·12-s + (−1.54 − 4.76i)13-s + (2.50 + 1.82i)14-s + (5.11 − 3.71i)15-s + (0.309 − 0.951i)16-s + (−2.39 + 7.37i)17-s + ⋯
L(s)  = 1  + (0.218 + 0.672i)2-s + (1.31 + 0.953i)3-s + (−0.404 + 0.293i)4-s + (0.310 − 0.957i)5-s + (−0.354 + 1.09i)6-s + (0.947 − 0.688i)7-s + (−0.286 − 0.207i)8-s + (0.504 + 1.55i)9-s + 0.711·10-s + (0.580 + 0.814i)11-s − 0.811·12-s + (−0.429 − 1.32i)13-s + (0.669 + 0.486i)14-s + (1.32 − 0.959i)15-s + (0.0772 − 0.237i)16-s + (−0.581 + 1.78i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 506 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 506 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.360 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(506\)    =    \(2 \cdot 11 \cdot 23\)
Sign: $0.360 - 0.932i$
Analytic conductor: \(4.04043\)
Root analytic conductor: \(2.01008\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{506} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 506,\ (\ :1/2),\ 0.360 - 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11186 + 1.44759i\)
\(L(\frac12)\) \(\approx\) \(2.11186 + 1.44759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (-1.92 - 2.70i)T \)
23 \( 1 + T \)
good3 \( 1 + (-2.27 - 1.65i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.695 + 2.14i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (-2.50 + 1.82i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (1.54 + 4.76i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.39 - 7.37i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (4.80 + 3.49i)T + (5.87 + 18.0i)T^{2} \)
29 \( 1 + (-3.88 + 2.82i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.461 - 1.41i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (7.51 - 5.46i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (3.10 + 2.25i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.79T + 43T^{2} \)
47 \( 1 + (3.74 + 2.72i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-0.190 - 0.584i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (5.65 - 4.10i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-0.273 + 0.843i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 1.43T + 67T^{2} \)
71 \( 1 + (-2.95 + 9.09i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-7.19 + 5.22i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-4.31 - 13.2i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.05 + 15.5i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (-2.52 - 7.77i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.60677909738087530957415729297, −10.11324311908879408566598024469, −8.991431307189385028168541655054, −8.410576747258395467089867539581, −7.88748947189129183714300170342, −6.55595093172408671766894331798, −4.93160097039055135821757802817, −4.57752687896507817852410601216, −3.56097508177225175490151167335, −1.87175298893089602496298833918, 1.79403496052115449506982614483, 2.42501792777697773821656865644, 3.44643519264449196739237246744, 4.83761607033920638553194630088, 6.41893692124462666399784621432, 7.04717432128844333240706775703, 8.317367629831217493652136768016, 8.857755212857811606560491707416, 9.697287828275952725373674521644, 10.96615296216814930714589703491

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