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Properties

Label 2-507-13.9-c1-0-1
Degree $2$
Conductor $507$
Sign $-0.611 + 0.791i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (−0.623 + 1.07i)2-s + (−0.5 + 0.866i)3-s + (0.222 + 0.385i)4-s − 2.80·5-s + (−0.623 − 1.07i)6-s + (2.40 + 4.15i)7-s − 3.04·8-s + (−0.499 − 0.866i)9-s + (1.74 − 3.02i)10-s + (−0.733 + 1.27i)11-s − 0.445·12-s − 5.98·14-s + (1.40 − 2.42i)15-s + (1.45 − 2.52i)16-s + (1.22 + 2.11i)17-s + 1.24·18-s + ⋯
L(s)  = 1  + (−0.440 + 0.763i)2-s + (−0.288 + 0.499i)3-s + (0.111 + 0.192i)4-s − 1.25·5-s + (−0.254 − 0.440i)6-s + (0.907 + 1.57i)7-s − 1.07·8-s + (−0.166 − 0.288i)9-s + (0.552 − 0.956i)10-s + (−0.221 + 0.383i)11-s − 0.128·12-s − 1.60·14-s + (0.361 − 0.626i)15-s + (0.363 − 0.630i)16-s + (0.296 + 0.513i)17-s + 0.293·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.611 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.611 + 0.791i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.611 + 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.244661 - 0.498394i\)
\(L(\frac12)\) \(\approx\) \(0.244661 - 0.498394i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (0.623 - 1.07i)T + (-1 - 1.73i)T^{2} \)
5 \( 1 + 2.80T + 5T^{2} \)
7 \( 1 + (-2.40 - 4.15i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.733 - 1.27i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.22 - 2.11i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.27 + 2.20i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.75 + 3.04i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.925 - 1.60i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.63T + 31T^{2} \)
37 \( 1 + (-2.27 + 3.94i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.623 - 1.07i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.19 + 2.06i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 + (1.08 + 1.88i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.91 - 6.78i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.79 + 3.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-4.41 - 7.65i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 7.69T + 73T^{2} \)
79 \( 1 + 4.02T + 79T^{2} \)
83 \( 1 + 0.652T + 83T^{2} \)
89 \( 1 + (3.14 - 5.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.01 - 8.68i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.46172124883636306979723020944, −10.82349032482150823382869775954, −9.267375663581040744267204723877, −8.668423377220578874580136551163, −7.973100353451714596104324609009, −7.16508200994173259069433388491, −5.93519573568424868358637514578, −5.06781234843590150447712330282, −3.87260986627731040037512834077, −2.52194600600372058720521402482, 0.40335543681034526766362535944, 1.54846859320585799697169659112, 3.30355837569467803352091487550, 4.32972556674601684195301237049, 5.60509209746762363061836574989, 7.00851883699688111577178816191, 7.64025696786318458345553413150, 8.398855880685278558323141172400, 9.702665482258327620211893590240, 10.75948152988387882526207236814

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