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Properties

Label 2-507-13.10-c1-0-26
Degree $2$
Conductor $507$
Sign $-0.996 - 0.0841i$
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  + (1.07 − 0.623i)2-s + (−0.5 − 0.866i)3-s + (−0.222 + 0.385i)4-s − 2.80i·5-s + (−1.07 − 0.623i)6-s + (−4.15 − 2.40i)7-s + 3.04i·8-s + (−0.499 + 0.866i)9-s + (−1.74 − 3.02i)10-s + (−1.27 + 0.733i)11-s + 0.445·12-s − 5.98·14-s + (−2.42 + 1.40i)15-s + (1.45 + 2.52i)16-s + (−1.22 + 2.11i)17-s + 1.24i·18-s + ⋯
L(s)  = 1  + (0.763 − 0.440i)2-s + (−0.288 − 0.499i)3-s + (−0.111 + 0.192i)4-s − 1.25i·5-s + (−0.440 − 0.254i)6-s + (−1.57 − 0.907i)7-s + 1.07i·8-s + (−0.166 + 0.288i)9-s + (−0.552 − 0.956i)10-s + (−0.383 + 0.221i)11-s + 0.128·12-s − 1.60·14-s + (−0.626 + 0.361i)15-s + (0.363 + 0.630i)16-s + (−0.296 + 0.513i)17-s + 0.293i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.996 - 0.0841i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.996 - 0.0841i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0341271 + 0.809995i\)
\(L(\frac12)\) \(\approx\) \(0.0341271 + 0.809995i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.07 + 0.623i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 2.80iT - 5T^{2} \)
7 \( 1 + (4.15 + 2.40i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.27 - 0.733i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (1.22 - 2.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.20 + 1.27i)T + (9.5 + 16.4i)T^{2} \)
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 \( 1 + 7.63iT - 31T^{2} \)
37 \( 1 + (-3.94 + 2.27i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.07 + 0.623i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.19 + 2.06i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.8iT - 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 + (-1.88 - 1.08i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.91 + 6.78i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.10 - 1.79i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-7.65 - 4.41i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.69iT - 73T^{2} \)
79 \( 1 + 4.02T + 79T^{2} \)
83 \( 1 + 0.652iT - 83T^{2} \)
89 \( 1 + (5.45 - 3.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-8.68 - 5.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.59729869552078908318986216066, −9.620806532350325489799469085561, −8.641558596714122362357219655233, −7.75623168263010866564369160035, −6.60817563018595204628830174620, −5.63195061049339482550334605027, −4.49168599406759350860458093938, −3.77056661561352096860819919455, −2.33599960684082878512451041543, −0.37034449921119987946255438498, 2.85119940877162239821246398307, 3.53272781393754365451363220863, 4.89482941618729772251283644567, 6.07136252398558104498640416294, 6.30663280108642489957909219143, 7.31812354288205105141540348057, 8.972036189036442456646302633904, 9.752298723013442265836713988262, 10.36799811216535513792756046320, 11.28681881026712450336213328027

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