Properties

Label 2-507-39.11-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  + (1.45 + 0.389i)2-s + (−0.866 − 1.5i)3-s + (0.232 + 0.133i)4-s + (2.90 − 2.90i)5-s + (−0.675 − 2.51i)6-s + (−1.84 − 1.84i)8-s + (−1.5 + 2.59i)9-s + (5.36 − 3.09i)10-s + (−0.285 + 1.06i)11-s − 0.464i·12-s + (−6.88 − 1.84i)15-s + (−2.23 − 3.86i)16-s + (−3.19 + 3.19i)18-s + (1.06 − 0.285i)20-s + (−0.830 + 1.43i)22-s + ⋯
L(s)  = 1  + (1.02 + 0.275i)2-s + (−0.499 − 0.866i)3-s + (0.116 + 0.0669i)4-s + (1.30 − 1.30i)5-s + (−0.275 − 1.02i)6-s + (−0.652 − 0.652i)8-s + (−0.5 + 0.866i)9-s + (1.69 − 0.979i)10-s + (−0.0860 + 0.321i)11-s − 0.133i·12-s + (−1.77 − 0.476i)15-s + (−0.558 − 0.966i)16-s + (−0.752 + 0.752i)18-s + (0.238 − 0.0638i)20-s + (−0.176 + 0.306i)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: $\chi_{507} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -0.0257 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48100 - 1.51969i\)
\(L(\frac12)\) \(\approx\) \(1.48100 - 1.51969i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.45 - 0.389i)T + (1.73 + i)T^{2} \)
5 \( 1 + (-2.90 + 2.90i)T - 5iT^{2} \)
7 \( 1 + (-6.06 + 3.5i)T^{2} \)
11 \( 1 + (0.285 - 1.06i)T + (-9.52 - 5.5i)T^{2} \)
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 \( 1 + (-9.79 - 2.62i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.59 - 6.59i)T + 47iT^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (14.8 - 3.97i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-6.92 + 12i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.27 + 4.75i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 - 73iT^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 + (9.29 - 9.29i)T - 83iT^{2} \)
89 \( 1 + (4.75 - 17.7i)T + (-77.0 - 44.5i)T^{2} \)
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

−10.81173966362980302365185598177, −9.640782064263691126445850641507, −9.004890051460996967445554082674, −7.82033564557144458729220294537, −6.54503631502113999127384956373, −5.84998381766898360600791289396, −5.20080855636045371637563059625, −4.38246814417981268494422555232, −2.42853375116889831060229058189, −1.03288671532338169759142507312, 2.46243456803328184106975842712, 3.33376386521510354470825606681, 4.41399924860492681751392714137, 5.72347183999337360156211713759, 5.88967740303572206829251448597, 7.11359361927102562440896111388, 8.808666967523502009295052632530, 9.590715751614117026633581052956, 10.53005391277418280781387535123, 11.01355032804935213648793694666

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