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Properties

Label 2-507-39.11-c1-0-8
Degree $2$
Conductor $507$
Sign $-0.477 - 0.878i$
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  + (0.866 + 1.5i)3-s + (−1.73 − i)4-s + (0.767 + 2.86i)7-s + (−1.5 + 2.59i)9-s − 3.46i·12-s + (1.99 + 3.46i)16-s + (−7.83 + 2.09i)19-s + (−3.63 + 3.63i)21-s + 5i·25-s − 5.19·27-s + (1.53 − 5.73i)28-s + (7.83 + 7.83i)31-s + (5.19 − 3i)36-s + (−2.09 − 0.562i)37-s + (−1.5 − 0.866i)43-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (−0.866 − 0.5i)4-s + (0.290 + 1.08i)7-s + (−0.5 + 0.866i)9-s − 0.999i·12-s + (0.499 + 0.866i)16-s + (−1.79 + 0.481i)19-s + (−0.792 + 0.792i)21-s + i·25-s − 1.00·27-s + (0.290 − 1.08i)28-s + (1.40 + 1.40i)31-s + (0.866 − 0.5i)36-s + (−0.344 − 0.0924i)37-s + (−0.228 − 0.132i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.477 - 0.878i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.571111 + 0.960378i\)
\(L(\frac12)\) \(\approx\) \(0.571111 + 0.960378i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 - 1.5i)T \)
13 \( 1 \)
good2 \( 1 + (1.73 + i)T^{2} \)
5 \( 1 - 5iT^{2} \)
7 \( 1 + (-0.767 - 2.86i)T + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (7.83 - 2.09i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.83 - 7.83i)T + 31iT^{2} \)
37 \( 1 + (2.09 + 0.562i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.5 + 0.866i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-4.33 + 7.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.205 - 0.767i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.36 + 9.36i)T - 73iT^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-16.4 + 4.40i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.88459121238058056674871440520, −10.25160565322746317750466618816, −9.311176196880634678871190305421, −8.675957360228361541742299441376, −8.118051625134217355869920439470, −6.35320255102913005744137941256, −5.32239260050040866946285818753, −4.63890012588682189311900963985, −3.50282289698192908640937145607, −2.05299892339382371345659382844, 0.64308601347123410262181001132, 2.44360027744202605357775370412, 3.85515500815609158984814522605, 4.59243241489671934249768490685, 6.21119680139012475574013448000, 7.12665778599877738550204658563, 8.083475485732295633049417322350, 8.518940272153428220526405798477, 9.607663078611278334427739642324, 10.55667298674133935607129504226

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