↑

Properties

Label 2-507-13.3-c1-0-22
Degree $2$
Conductor $507$
Sign $0.522 + 0.852i$
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  + (0.207 + 0.358i)2-s + (−0.5 − 0.866i)3-s + (0.914 − 1.58i)4-s + 2.82·5-s + (0.207 − 0.358i)6-s + (1.41 − 2.44i)7-s + 1.58·8-s + (−0.499 + 0.866i)9-s + (0.585 + 1.01i)10-s + (−1 − 1.73i)11-s − 1.82·12-s + 1.17·14-s + (−1.41 − 2.44i)15-s + (−1.49 − 2.59i)16-s + (−3.82 + 6.63i)17-s − 0.414·18-s + ⋯
L(s)  = 1  + (0.146 + 0.253i)2-s + (−0.288 − 0.499i)3-s + (0.457 − 0.791i)4-s + 1.26·5-s + (0.0845 − 0.146i)6-s + (0.534 − 0.925i)7-s + 0.560·8-s + (−0.166 + 0.288i)9-s + (0.185 + 0.320i)10-s + (−0.301 − 0.522i)11-s − 0.527·12-s + 0.313·14-s + (−0.365 − 0.632i)15-s + (−0.374 − 0.649i)16-s + (−0.928 + 1.60i)17-s − 0.0976·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.522 + 0.852i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (484, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.522 + 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70119 - 0.953309i\)
\(L(\frac12)\) \(\approx\) \(1.70119 - 0.953309i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-0.207 - 0.358i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 - 2.82T + 5T^{2} \)
7 \( 1 + (-1.41 + 2.44i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (3.82 - 6.63i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.41 - 2.44i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1 + 1.73i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.17T + 31T^{2} \)
37 \( 1 + (3.82 + 6.63i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.58 - 4.47i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.828 + 1.43i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.6T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + (-3.82 + 6.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.65 - 11.5i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.41 - 5.91i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1 + 1.73i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 0.343T + 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 3.65T + 83T^{2} \)
89 \( 1 + (-7.41 - 12.8i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-1.82 + 3.16i)T + (-48.5 - 84.0i)T^{2} \)
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

−10.65433315101747545523567762092, −10.23297020961379057939477395500, −9.029087794000838497663895177775, −7.85367015335727319243938035844, −6.90616231386214811538285367320, −6.00879206887661471875555171306, −5.54158196755666875524205763952, −4.21130956263690406268950429095, −2.20169886619449693510770836419, −1.30196099758597099383030985737, 2.12457780496368629260633669199, 2.79537479818816027185796574490, 4.54936766426418885587065903167, 5.25187016563718318674416825070, 6.40940494951528531770607497376, 7.29809213740979468354591257644, 8.680446717925166289506703244327, 9.226798472947713878655754622885, 10.29127628754537152267862069135, 11.10471092894043419211742644436

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