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Properties

Label 2-507-13.4-c1-0-23
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  + (2.04 + 1.17i)2-s + (0.5 − 0.866i)3-s + (1.77 + 3.07i)4-s − 3.69i·5-s + (2.04 − 1.17i)6-s + (−0.694 + 0.400i)7-s + 3.66i·8-s + (−0.499 − 0.866i)9-s + (4.35 − 7.53i)10-s + (2.46 + 1.42i)11-s + 3.55·12-s − 1.89·14-s + (−3.19 − 1.84i)15-s + (−0.763 + 1.32i)16-s + (1.46 + 2.54i)17-s − 2.35i·18-s + ⋯
L(s)  = 1  + (1.44 + 0.833i)2-s + (0.288 − 0.499i)3-s + (0.888 + 1.53i)4-s − 1.65i·5-s + (0.833 − 0.481i)6-s + (−0.262 + 0.151i)7-s + 1.29i·8-s + (−0.166 − 0.288i)9-s + (1.37 − 2.38i)10-s + (0.744 + 0.429i)11-s + 1.02·12-s − 0.505·14-s + (−0.825 − 0.476i)15-s + (−0.190 + 0.330i)16-s + (0.356 + 0.617i)17-s − 0.555i·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: $\chi_{507} (316, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.996 - 0.0841i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.31096 + 0.139499i\)
\(L(\frac12)\) \(\approx\) \(3.31096 + 0.139499i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.04 - 1.17i)T + (1 + 1.73i)T^{2} \)
5 \( 1 + 3.69iT - 5T^{2} \)
7 \( 1 + (0.694 - 0.400i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2.46 - 1.42i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.46 - 2.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.11 + 1.22i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.89 - 6.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.92 - 3.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 2.34iT - 31T^{2} \)
37 \( 1 + (6.44 + 3.72i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.736 + 0.425i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.807 + 1.39i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 2.44iT - 47T^{2} \)
53 \( 1 + 9.96T + 53T^{2} \)
59 \( 1 + (4.66 - 2.69i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.62 - 11.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-12.4 - 7.19i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (7.03 - 4.06i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 11.8iT - 73T^{2} \)
79 \( 1 - 5.40T + 79T^{2} \)
83 \( 1 + 7.04iT - 83T^{2} \)
89 \( 1 + (0.980 + 0.565i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.14 + 2.97i)T + (48.5 - 84.0i)T^{2} \)
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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

−11.58838799644635082609046048045, −9.712059640600929111596464608070, −8.940077547664651387949544091345, −7.942362985821669748274925297140, −7.17084986954701494288709421077, −6.00988271836912080608062980198, −5.36194153102611570943590925705, −4.34714313253602073386335832541, −3.46925682382829106634744415211, −1.58161297704391091233768132710, 2.20783128811651261385230485681, 3.26593760233507623285521704922, 3.70293338717213261031487298554, 4.95979822586711789756101834366, 6.18042851142834405557312790927, 6.77581259493402226903442222609, 8.143334718093753780068007887963, 9.693826139489641877389563014014, 10.29251272335874431922792787521, 11.08243331689105393089546487238

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