↑

Properties

Label 2-507-39.32-c1-0-39
Degree $2$
Conductor $507$
Sign $0.0257 + 0.999i$
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  + (2.31 − 0.619i)2-s + (0.866 − 1.5i)3-s + (3.23 − 1.86i)4-s + (−1.23 − 1.23i)5-s + (1.07 − 4.00i)6-s + (2.93 − 2.93i)8-s + (−1.5 − 2.59i)9-s + (−3.63 − 2.09i)10-s + (1.69 + 6.31i)11-s − 6.46i·12-s + (−2.93 + 0.785i)15-s + (1.23 − 2.13i)16-s + (−5.07 − 5.07i)18-s + (−6.31 − 1.69i)20-s + (7.83 + 13.5i)22-s + ⋯
L(s)  = 1  + (1.63 − 0.438i)2-s + (0.499 − 0.866i)3-s + (1.61 − 0.933i)4-s + (−0.554 − 0.554i)5-s + (0.438 − 1.63i)6-s + (1.03 − 1.03i)8-s + (−0.5 − 0.866i)9-s + (−1.14 − 0.663i)10-s + (0.510 + 1.90i)11-s − 1.86i·12-s + (−0.757 + 0.202i)15-s + (0.308 − 0.533i)16-s + (−1.19 − 1.19i)18-s + (−1.41 − 0.378i)20-s + (1.66 + 2.89i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0257 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.0257 + 0.999i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (188, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.0257 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.63480 - 2.56771i\)
\(L(\frac12)\) \(\approx\) \(2.63480 - 2.56771i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.31 + 0.619i)T + (1.73 - i)T^{2} \)
5 \( 1 + (1.23 + 1.23i)T + 5iT^{2} \)
7 \( 1 + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (-1.69 - 6.31i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 31iT^{2} \)
37 \( 1 + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-7.55 + 2.02i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-3.46 + 2i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (7.10 - 7.10i)T - 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-0.453 - 0.121i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.92 + 12i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (4.17 - 15.5i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + 73iT^{2} \)
79 \( 1 + 10.3T + 79T^{2} \)
83 \( 1 + (8.91 + 8.91i)T + 83iT^{2} \)
89 \( 1 + (1.11 + 4.17i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (84.0 + 48.5i)T^{2} \)
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

−11.24995224188582753994626894837, −9.913107181642176639511117609924, −8.865994799991644726523298652083, −7.65196693298823977001002321826, −6.91066093628545263928937303087, −5.92283952081654133871477847041, −4.64173271410630177290949721004, −4.03031172764171224342283785422, −2.69961439291851440483029780046, −1.61952475841461065483550852555, 2.83894836680697023088072432065, 3.51062001394689198702561074938, 4.24515812705287183770135197555, 5.43986292700352144443774142421, 6.17552889206538220450024168751, 7.31729298990015027633031830552, 8.285946222846588377076207628120, 9.242850461847932755091505336326, 10.65672153698792997598191768812, 11.28430140420076721228729271090

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