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Properties

Label 2-507-39.5-c1-0-19
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

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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: $\chi_{507} (239, \cdot )$
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 \( 1 - 97iT^{2} \)
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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

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

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