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Properties

Label 2-507-13.3-c1-0-7
Degree $2$
Conductor $507$
Sign $-0.522 - 0.852i$
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  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (0.500 − 0.866i)4-s − 2·5-s + (−0.499 + 0.866i)6-s + (−2 + 3.46i)7-s + 3·8-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)10-s + (2 + 3.46i)11-s + 12-s − 3.99·14-s + (−1 − 1.73i)15-s + (0.500 + 0.866i)16-s + (−1 + 1.73i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (0.250 − 0.433i)4-s − 0.894·5-s + (−0.204 + 0.353i)6-s + (−0.755 + 1.30i)7-s + 1.06·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s + (0.603 + 1.04i)11-s + 0.288·12-s − 1.06·14-s + (−0.258 − 0.447i)15-s + (0.125 + 0.216i)16-s + (−0.242 + 0.420i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776549 + 1.38576i\)
\(L(\frac12)\) \(\approx\) \(0.776549 + 1.38576i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 2T + 5T^{2} \)
7 \( 1 + (2 - 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-2 - 3.46i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5 - 8.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6 + 10.3i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (-6 + 10.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (1 + 1.73i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5 + 8.66i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.20364562420934786870735072249, −10.25070766310604464622937920554, −9.356319336327605337714852632156, −8.584221031990018921512585595383, −7.41413062405619755433591484092, −6.60603110459454829388153230802, −5.61031594342196064957574046837, −4.65206823403212734468833470923, −3.58577020098926599084787960698, −2.12317333635998849505750016637, 0.839282436912720068322739163135, 2.73303667353826011515671348488, 3.73668906108316903959501848053, 4.23575502597630640115344810438, 6.19417251910214000081290352133, 7.16848144627570725278851980456, 7.72219644707522635571244564930, 8.675479495880407507416020894278, 9.929326569523744335718636515994, 10.91082762085108215318468509349

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