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Properties

Label 2-507-13.3-c1-0-3
Degree $2$
Conductor $507$
Sign $-0.872 + 0.488i$
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  + (1.28 + 2.21i)2-s + (−0.5 − 0.866i)3-s + (−2.28 + 3.95i)4-s − 0.561·5-s + (1.28 − 2.21i)6-s + (−1.78 + 3.08i)7-s − 6.56·8-s + (−0.499 + 0.866i)9-s + (−0.719 − 1.24i)10-s + (−1 − 1.73i)11-s + 4.56·12-s − 9.12·14-s + (0.280 + 0.486i)15-s + (−3.84 − 6.65i)16-s + (−1.28 + 2.21i)17-s − 2.56·18-s + ⋯
L(s)  = 1  + (0.905 + 1.56i)2-s + (−0.288 − 0.499i)3-s + (−1.14 + 1.97i)4-s − 0.251·5-s + (0.522 − 0.905i)6-s + (−0.673 + 1.16i)7-s − 2.31·8-s + (−0.166 + 0.288i)9-s + (−0.227 − 0.393i)10-s + (−0.301 − 0.522i)11-s + 1.31·12-s − 2.43·14-s + (0.0724 + 0.125i)15-s + (−0.960 − 1.66i)16-s + (−0.310 + 0.538i)17-s − 0.603·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.872 + 0.488i$
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: \((2,\ 507,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304754 - 1.16724i\)
\(L(\frac12)\) \(\approx\) \(0.304754 - 1.16724i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.28 - 2.21i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + 0.561T + 5T^{2} \)
7 \( 1 + (1.78 - 3.08i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.28 - 2.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.561 - 0.972i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.84 - 4.92i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (-1.71 - 2.97i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-1.28 - 2.21i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.219 - 0.379i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.24T + 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + (5.56 - 9.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.06 - 10.4i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.219 - 0.379i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 1.87T + 73T^{2} \)
79 \( 1 - 9.56T + 79T^{2} \)
83 \( 1 - 9.12T + 83T^{2} \)
89 \( 1 + (-6.56 - 11.3i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (2.21 - 3.84i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.10886074694302035525209266262, −10.63692158689389082826484948782, −9.137941449967986681161789006876, −8.405938749406118572164104092231, −7.63661169518696744267818112828, −6.54295540324409244938946421240, −6.00364067197508502832458875691, −5.25738935679637812614144233208, −4.01362408826395669446187411039, −2.74794410724370095618615346792, 0.54296060004910128312835454682, 2.38450978439024155979693654599, 3.66302251458454301614862436545, 4.23115734252547895190767245394, 5.18508400443900215780810535581, 6.37901254865646015753273483190, 7.61643638444700356476380948937, 9.260354150233821760845694221966, 9.947290174622120162720620225379, 10.51837116571101508894078048619

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