↑

Properties

Label 2-300-100.27-c1-0-10
Degree $2$
Conductor $300$
Sign $0.481 - 0.876i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
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  + (−0.893 + 1.09i)2-s + (−0.891 + 0.453i)3-s + (−0.404 − 1.95i)4-s + (0.968 + 2.01i)5-s + (0.297 − 1.38i)6-s + (2.67 − 2.67i)7-s + (2.50 + 1.30i)8-s + (0.587 − 0.809i)9-s + (−3.07 − 0.738i)10-s + (−1.87 − 2.57i)11-s + (1.24 + 1.56i)12-s + (5.65 − 0.895i)13-s + (0.544 + 5.32i)14-s + (−1.77 − 1.35i)15-s + (−3.67 + 1.58i)16-s + (−2.03 + 3.99i)17-s + ⋯
L(s)  = 1  + (−0.631 + 0.775i)2-s + (−0.514 + 0.262i)3-s + (−0.202 − 0.979i)4-s + (0.433 + 0.901i)5-s + (0.121 − 0.564i)6-s + (1.01 − 1.01i)7-s + (0.887 + 0.461i)8-s + (0.195 − 0.269i)9-s + (−0.972 − 0.233i)10-s + (−0.564 − 0.777i)11-s + (0.360 + 0.450i)12-s + (1.56 − 0.248i)13-s + (0.145 + 1.42i)14-s + (−0.459 − 0.350i)15-s + (−0.918 + 0.396i)16-s + (−0.494 + 0.969i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.481 - 0.876i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.833783 + 0.493035i\)
\(L(\frac12)\) \(\approx\) \(0.833783 + 0.493035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.893 - 1.09i)T \)
3 \( 1 + (0.891 - 0.453i)T \)
5 \( 1 + (-0.968 - 2.01i)T \)
good7 \( 1 + (-2.67 + 2.67i)T - 7iT^{2} \)
11 \( 1 + (1.87 + 2.57i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-5.65 + 0.895i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (2.03 - 3.99i)T + (-9.99 - 13.7i)T^{2} \)
19 \( 1 + (-0.869 - 2.67i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-5.81 - 0.921i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-8.67 - 2.81i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (6.80 - 2.21i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.906 + 5.72i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (6.17 + 4.48i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 + (-4.72 - 4.72i)T + 43iT^{2} \)
47 \( 1 + (2.06 + 4.04i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-1.95 - 3.83i)T + (-31.1 + 42.8i)T^{2} \)
59 \( 1 + (5.25 + 3.81i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.90 + 2.84i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-8.86 - 4.51i)T + (39.3 + 54.2i)T^{2} \)
71 \( 1 + (2.55 + 0.831i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.818 + 5.16i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-1.62 + 4.99i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-0.853 + 1.67i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (6.51 + 8.96i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (4.14 - 2.11i)T + (57.0 - 78.4i)T^{2} \)
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

−11.19581029218068934379651683641, −10.74267191928686639058668335974, −10.40856387004331916206319790596, −8.894387849792179794391710215016, −8.040636598887173460736614019493, −6.99831307795923407143887861581, −6.10431379380069017718686146878, −5.19276750655431284264483769233, −3.71728165176429032104993541858, −1.34991871673045066707643256709, 1.26324410916120045998547220169, 2.47529202889858733656677151121, 4.56289835008338865808023866392, 5.29379330682612349394955105465, 6.80671341346427303240829437135, 8.170658411370902268676807429066, 8.774230330922605559392546452856, 9.623638433920122083694825735550, 10.88489282819865195456729621814, 11.52561635370883365395710948523

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