Properties

Label 2-507-39.5-c1-0-35
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  − 1.73·3-s − 2i·4-s + (−2.09 − 2.09i)7-s + 2.99·9-s + 3.46i·12-s − 4·16-s + (−5.73 + 5.73i)19-s + (3.63 + 3.63i)21-s − 5i·25-s − 5.19·27-s + (−4.19 + 4.19i)28-s + (−7.83 + 7.83i)31-s − 5.99i·36-s + (−1.53 − 1.53i)37-s − 1.73i·43-s + ⋯
L(s)  = 1  − 1.00·3-s i·4-s + (−0.792 − 0.792i)7-s + 0.999·9-s + 0.999i·12-s − 16-s + (−1.31 + 1.31i)19-s + (0.792 + 0.792i)21-s i·25-s − 1.00·27-s + (−0.792 + 0.792i)28-s + (−1.40 + 1.40i)31-s − 0.999i·36-s + (−0.252 − 0.252i)37-s − 0.264i·43-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

\(L(1)\) \(\approx\) \(0.0353849 + 0.238976i\)
\(L(\frac12)\) \(\approx\) \(0.0353849 + 0.238976i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT^{2} \)
5 \( 1 + 5iT^{2} \)
7 \( 1 + (2.09 + 2.09i)T + 7iT^{2} \)
11 \( 1 - 11iT^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + (5.73 - 5.73i)T - 19iT^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (7.83 - 7.83i)T - 31iT^{2} \)
37 \( 1 + (1.53 + 1.53i)T + 37iT^{2} \)
41 \( 1 + 41iT^{2} \)
43 \( 1 + 1.73iT - 43T^{2} \)
47 \( 1 - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 + 8.66T + 61T^{2} \)
67 \( 1 + (-0.562 + 0.562i)T - 67iT^{2} \)
71 \( 1 + 71iT^{2} \)
73 \( 1 + (9.36 + 9.36i)T + 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 - 89iT^{2} \)
97 \( 1 + (-12.0 + 12.0i)T - 97iT^{2} \)
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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

−10.41950554714433926610856258166, −9.977770093014047931327513518199, −8.834652644040702220137722865503, −7.37004458763722651040887815706, −6.49141196469289806162188124450, −5.91949938385819876397981869923, −4.80105152107864745881265526075, −3.79268767175375804761478461030, −1.70367055843118625946056654726, −0.15698818360259769906684857039, 2.35783233922793297285974502560, 3.68982284830079010331445129948, 4.79241584067534060575735726657, 5.94850236425707780040984914186, 6.76835648053072734459229328095, 7.62144378737474454015191852309, 8.894683782072345328249586496779, 9.476773999613388570092562154342, 10.78556990362931484510684993244, 11.43538467157131358612859659223

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