↑

Properties

Label 2-507-39.8-c1-0-5
Degree $2$
Conductor $507$
Sign $-0.957 + 0.289i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.69 + 1.69i)2-s − 1.73·3-s − 3.73i·4-s + (−1.23 + 1.23i)5-s + (2.93 − 2.93i)6-s + (2.93 + 2.93i)8-s + 2.99·9-s − 4.19i·10-s + (4.62 + 4.62i)11-s + 6.46i·12-s + (2.14 − 2.14i)15-s − 2.46·16-s + (−5.07 + 5.07i)18-s + (4.62 + 4.62i)20-s − 15.6·22-s + ⋯
L(s)  = 1  + (−1.19 + 1.19i)2-s − 1.00·3-s − 1.86i·4-s + (−0.554 + 0.554i)5-s + (1.19 − 1.19i)6-s + (1.03 + 1.03i)8-s + 0.999·9-s − 1.32i·10-s + (1.39 + 1.39i)11-s + 1.86i·12-s + (0.554 − 0.554i)15-s − 0.616·16-s + (−1.19 + 1.19i)18-s + (1.03 + 1.03i)20-s − 3.33·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.957 + 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.957 + 0.289i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.957 + 0.289i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0524409 - 0.354166i\)
\(L(\frac12)\) \(\approx\) \(0.0524409 - 0.354166i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73T \)
13 \( 1 \)
good2 \( 1 + (1.69 - 1.69i)T - 2iT^{2} \)
5 \( 1 + (1.23 - 1.23i)T - 5iT^{2} \)
7 \( 1 - 7iT^{2} \)
11 \( 1 + (-4.62 - 4.62i)T + 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31iT^{2} \)
37 \( 1 - 37iT^{2} \)
41 \( 1 + (5.53 - 5.53i)T - 41iT^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 + (7.10 + 7.10i)T + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (0.332 + 0.332i)T + 59iT^{2} \)
61 \( 1 - 13.8T + 61T^{2} \)
67 \( 1 + 67iT^{2} \)
71 \( 1 + (11.3 - 11.3i)T - 71iT^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (8.91 - 8.91i)T - 83iT^{2} \)
89 \( 1 + (3.05 + 3.05i)T + 89iT^{2} \)
97 \( 1 + 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.28226484144919312067932750781, −10.14360053890275136933341888512, −9.695153232119295266080528387262, −8.651598694586480859741007533325, −7.45085446184231745133837866199, −6.96235816735254795831297167934, −6.34171707869110473580380117646, −5.16508319906434781603128638315, −3.98152206935158859395202247106, −1.43933373722273104205715592922, 0.41867206179546221929824922636, 1.47998144922730687466026414876, 3.37124666151352713988568052088, 4.33789755386919939016430319827, 5.80308974991411047295512309273, 6.90096393378889863103559155319, 8.153914594461172768075226261279, 8.795127409385298445395348240656, 9.670151954331657137084752914666, 10.55224015262241203045291256931

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