↑

Properties

Label 2-252-36.23-c1-0-8
Degree $2$
Conductor $252$
Sign $0.390 - 0.920i$
Analytic cond. $2.01223$
Root an. cond. $1.41853$
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  + (−1.40 − 0.171i)2-s + (−1.10 + 1.33i)3-s + (1.94 + 0.481i)4-s + (3.64 + 2.10i)5-s + (1.77 − 1.68i)6-s + (0.866 − 0.5i)7-s + (−2.64 − 1.00i)8-s + (−0.568 − 2.94i)9-s + (−4.75 − 3.57i)10-s + (1.11 + 1.92i)11-s + (−2.78 + 2.06i)12-s + (1.14 − 1.98i)13-s + (−1.30 + 0.553i)14-s + (−6.82 + 2.54i)15-s + (3.53 + 1.86i)16-s + 3.74i·17-s + ⋯
L(s)  = 1  + (−0.992 − 0.121i)2-s + (−0.636 + 0.771i)3-s + (0.970 + 0.240i)4-s + (1.62 + 0.940i)5-s + (0.725 − 0.688i)6-s + (0.327 − 0.188i)7-s + (−0.934 − 0.356i)8-s + (−0.189 − 0.981i)9-s + (−1.50 − 1.13i)10-s + (0.335 + 0.581i)11-s + (−0.803 + 0.595i)12-s + (0.317 − 0.550i)13-s + (−0.347 + 0.147i)14-s + (−1.76 + 0.657i)15-s + (0.884 + 0.467i)16-s + 0.907i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(252\)    =    \(2^{2} \cdot 3^{2} \cdot 7\)
Sign: $0.390 - 0.920i$
Analytic conductor: \(2.01223\)
Root analytic conductor: \(1.41853\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{252} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 252,\ (\ :1/2),\ 0.390 - 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750851 + 0.496841i\)
\(L(\frac12)\) \(\approx\) \(0.750851 + 0.496841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 + 0.171i)T \)
3 \( 1 + (1.10 - 1.33i)T \)
7 \( 1 + (-0.866 + 0.5i)T \)
good5 \( 1 + (-3.64 - 2.10i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.11 - 1.92i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.14 + 1.98i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.74iT - 17T^{2} \)
19 \( 1 + 4.90iT - 19T^{2} \)
23 \( 1 + (2.71 - 4.70i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.985 + 0.568i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-1.45 - 0.837i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 11.8T + 37T^{2} \)
41 \( 1 + (2.39 + 1.38i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.416 + 0.240i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.47 + 2.55i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 4.61iT - 53T^{2} \)
59 \( 1 + (-1.12 + 1.94i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.35 + 4.07i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.4 - 6.63i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.00T + 71T^{2} \)
73 \( 1 + 5.14T + 73T^{2} \)
79 \( 1 + (-11.6 + 6.72i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.65 + 11.5i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 12.3iT - 89T^{2} \)
97 \( 1 + (-2.52 - 4.37i)T + (-48.5 + 84.0i)T^{2} \)
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

−11.77571646256251933792989113975, −10.83342458919770101703228161927, −10.27621967411656535257299930172, −9.662594519529651684593770264827, −8.710869484547294297915984825352, −7.07113977800575895168009284655, −6.30062044962800751434128722118, −5.33829513512438230774803784699, −3.37278738391870140906307594233, −1.81470718931605519380196840773, 1.20111497235519494699376623032, 2.21029006128821656884329680922, 5.13956853206531407003774140794, 5.99628427520446831812880591129, 6.70876972381052145257824224353, 8.188328049431473617463544030709, 8.888790262388604756341581744913, 9.888117719250362135545689735195, 10.76279024966496549089037572343, 11.91282749572357319548692625496

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