↑

Properties

Label 2-504-63.38-c1-0-7
Degree $2$
Conductor $504$
Sign $-0.869 - 0.493i$
Analytic cond. $4.02446$
Root an. cond. $2.00610$
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  + (0.0250 + 1.73i)3-s + (−1.42 + 2.46i)5-s + (1.38 + 2.25i)7-s + (−2.99 + 0.0869i)9-s + (2.12 − 1.22i)11-s + (3.06 − 1.76i)13-s + (−4.30 − 2.40i)15-s + (−2.91 + 5.05i)17-s + (−2.90 + 1.67i)19-s + (−3.87 + 2.45i)21-s + (−6.94 − 4.00i)23-s + (−1.55 − 2.69i)25-s + (−0.225 − 5.19i)27-s + (−1.45 − 0.839i)29-s − 3.98i·31-s + ⋯
L(s)  = 1  + (0.0144 + 0.999i)3-s + (−0.636 + 1.10i)5-s + (0.522 + 0.852i)7-s + (−0.999 + 0.0289i)9-s + (0.639 − 0.369i)11-s + (0.848 − 0.490i)13-s + (−1.11 − 0.620i)15-s + (−0.708 + 1.22i)17-s + (−0.665 + 0.384i)19-s + (−0.844 + 0.535i)21-s + (−1.44 − 0.835i)23-s + (−0.310 − 0.538i)25-s + (−0.0434 − 0.999i)27-s + (−0.269 − 0.155i)29-s − 0.715i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.869 - 0.493i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(504\)    =    \(2^{3} \cdot 3^{2} \cdot 7\)
Sign: $-0.869 - 0.493i$
Analytic conductor: \(4.02446\)
Root analytic conductor: \(2.00610\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{504} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 504,\ (\ :1/2),\ -0.869 - 0.493i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.292449 + 1.10878i\)
\(L(\frac12)\) \(\approx\) \(0.292449 + 1.10878i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.0250 - 1.73i)T \)
7 \( 1 + (-1.38 - 2.25i)T \)
good5 \( 1 + (1.42 - 2.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-2.12 + 1.22i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.06 + 1.76i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.91 - 5.05i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.90 - 1.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.94 + 4.00i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.45 + 0.839i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.98iT - 31T^{2} \)
37 \( 1 + (-4.07 - 7.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5.43 - 9.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.27 + 5.67i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6.62T + 47T^{2} \)
53 \( 1 + (7.64 + 4.41i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 0.356T + 59T^{2} \)
61 \( 1 + 2.91iT - 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 9.96iT - 71T^{2} \)
73 \( 1 + (-5.42 - 3.13i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 + (0.189 - 0.327i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.05 - 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.00 - 2.31i)T + (48.5 + 84.0i)T^{2} \)
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

−11.12770667141783996178611996796, −10.63047823316347789881844983030, −9.610770391712907401037824934404, −8.392108811241516359736751394029, −8.149949811604927955515074802321, −6.36368018324961503815234235369, −5.91014672964534777639860289960, −4.32531556357141358215178585544, −3.64474417010211778152884527455, −2.39207279347846208463970150745, 0.69859756357091885650189235210, 1.92866502128257266557225246594, 3.88607086552367881685153050330, 4.67423263251056467572489605484, 6.00070458854987943089356553372, 7.11275016716060557027603335465, 7.73906266999541540837411966188, 8.706917169664419654693696448739, 9.301908652310233154769345831578, 10.91996308936816807286594564584

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