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Properties

Label 2-507-507.233-c0-0-0
Degree $2$
Conductor $507$
Sign $0.848 + 0.529i$
Analytic cond. $0.253025$
Root an. cond. $0.503016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.748 − 0.663i)3-s + (0.970 + 0.239i)4-s + (−1.24 − 0.470i)7-s + (0.120 − 0.992i)9-s + (0.885 − 0.464i)12-s − 13-s + (0.885 + 0.464i)16-s + 1.98i·19-s + (−1.24 + 0.470i)21-s + (−0.568 + 0.822i)25-s + (−0.568 − 0.822i)27-s + (−1.09 − 0.753i)28-s + (−0.393 + 0.271i)31-s + (0.354 − 0.935i)36-s + (1.53 − 1.06i)37-s + ⋯
L(s)  = 1  + (0.748 − 0.663i)3-s + (0.970 + 0.239i)4-s + (−1.24 − 0.470i)7-s + (0.120 − 0.992i)9-s + (0.885 − 0.464i)12-s − 13-s + (0.885 + 0.464i)16-s + 1.98i·19-s + (−1.24 + 0.470i)21-s + (−0.568 + 0.822i)25-s + (−0.568 − 0.822i)27-s + (−1.09 − 0.753i)28-s + (−0.393 + 0.271i)31-s + (0.354 − 0.935i)36-s + (1.53 − 1.06i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.848 + 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.165820456\)
\(L(\frac12)\) \(\approx\) \(1.165820456\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.748 + 0.663i)T \)
13 \( 1 + T \)
good2 \( 1 + (-0.970 - 0.239i)T^{2} \)
5 \( 1 + (0.568 - 0.822i)T^{2} \)
7 \( 1 + (1.24 + 0.470i)T + (0.748 + 0.663i)T^{2} \)
11 \( 1 + (-0.970 + 0.239i)T^{2} \)
17 \( 1 + (0.748 + 0.663i)T^{2} \)
19 \( 1 - 1.98iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.970 + 0.239i)T^{2} \)
31 \( 1 + (0.393 - 0.271i)T + (0.354 - 0.935i)T^{2} \)
37 \( 1 + (-1.53 + 1.06i)T + (0.354 - 0.935i)T^{2} \)
41 \( 1 + (0.120 - 0.992i)T^{2} \)
43 \( 1 + (0.402 - 0.583i)T + (-0.354 - 0.935i)T^{2} \)
47 \( 1 + (0.885 - 0.464i)T^{2} \)
53 \( 1 + (0.748 + 0.663i)T^{2} \)
59 \( 1 + (0.568 - 0.822i)T^{2} \)
61 \( 1 + (0.688 + 1.81i)T + (-0.748 + 0.663i)T^{2} \)
67 \( 1 + (0.393 + 1.59i)T + (-0.885 + 0.464i)T^{2} \)
71 \( 1 + (0.120 - 0.992i)T^{2} \)
73 \( 1 + (-0.922 + 0.112i)T + (0.970 - 0.239i)T^{2} \)
79 \( 1 + (1.45 - 0.358i)T + (0.885 - 0.464i)T^{2} \)
83 \( 1 + (0.120 + 0.992i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (-0.568 - 0.822i)T^{2} \)
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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

−11.08549241500513816634178527605, −9.956885000328671524964942829285, −9.459939037926284948286755240700, −7.997137241409117435435251869888, −7.49495091010362168716454944000, −6.61789864049512271989089226094, −5.86188775446798106219315658845, −3.82202112636565589607543718690, −3.05615725467250002252820950110, −1.83803678368782093799292544924, 2.47169652748095483524055461871, 2.95498006092252172312435490959, 4.44761121003268608269094844984, 5.65584521556781260151982711004, 6.72747653786229142100371340864, 7.49810410913055399763180087339, 8.707370974350212497522289179238, 9.636631724573293951697107074636, 10.06636474001072829287626208194, 11.12058665318409506545852299562

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