↑

Properties

Label 2-24e2-144.13-c1-0-1
Degree $2$
Conductor $576$
Sign $-0.941 + 0.335i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
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  + (−0.388 + 1.68i)3-s + (0.226 + 0.846i)5-s + (−0.567 + 0.327i)7-s + (−2.69 − 1.31i)9-s + (−5.75 − 1.54i)11-s + (−4.44 + 1.19i)13-s + (−1.51 + 0.0535i)15-s + 2.75·17-s + (1.73 + 1.73i)19-s + (−0.332 − 1.08i)21-s + (−3.50 − 2.02i)23-s + (3.66 − 2.11i)25-s + (3.26 − 4.04i)27-s + (−0.662 + 2.47i)29-s + (−2.08 + 3.61i)31-s + ⋯
L(s)  = 1  + (−0.224 + 0.974i)3-s + (0.101 + 0.378i)5-s + (−0.214 + 0.123i)7-s + (−0.899 − 0.437i)9-s + (−1.73 − 0.464i)11-s + (−1.23 + 0.330i)13-s + (−0.391 + 0.0138i)15-s + 0.668·17-s + (0.398 + 0.398i)19-s + (−0.0724 − 0.236i)21-s + (−0.731 − 0.422i)23-s + (0.732 − 0.423i)25-s + (0.628 − 0.777i)27-s + (−0.123 + 0.459i)29-s + (−0.375 + 0.649i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.941 + 0.335i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.941 + 0.335i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0605457 - 0.349936i\)
\(L(\frac12)\) \(\approx\) \(0.0605457 - 0.349936i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.388 - 1.68i)T \)
good5 \( 1 + (-0.226 - 0.846i)T + (-4.33 + 2.5i)T^{2} \)
7 \( 1 + (0.567 - 0.327i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.75 + 1.54i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.44 - 1.19i)T + (11.2 - 6.5i)T^{2} \)
17 \( 1 - 2.75T + 17T^{2} \)
19 \( 1 + (-1.73 - 1.73i)T + 19iT^{2} \)
23 \( 1 + (3.50 + 2.02i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.662 - 2.47i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (2.08 - 3.61i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.30 - 4.30i)T - 37iT^{2} \)
41 \( 1 + (6.15 + 3.55i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.841 - 0.225i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (4.65 + 8.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (7.64 - 7.64i)T - 53iT^{2} \)
59 \( 1 + (-1.83 - 6.83i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.01 + 3.77i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (-11.6 + 3.11i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 4.34iT - 71T^{2} \)
73 \( 1 - 0.656iT - 73T^{2} \)
79 \( 1 + (-8.16 - 14.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.43 + 5.36i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + 5.11iT - 89T^{2} \)
97 \( 1 + (-3.05 - 5.29i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.91934029214518342466828289653, −10.26652900186153120604784915946, −9.791575976669442215743120840407, −8.627593257060424060879456573863, −7.76513216875685638013629666754, −6.61133834000646041402757837871, −5.42266074817107165402899971022, −4.91986687377135305477727967319, −3.43964088846420111276581314606, −2.57451911956199753533507218590, 0.18865654597759925113457780094, 2.01347504531602275246742423120, 3.08478167989973949234686985031, 5.01353452573557741637440929601, 5.43638443597646575927134567463, 6.75931222587245181270788543168, 7.69030405427670342738751707152, 8.054533583772852783338940693590, 9.481854135148594992606221786962, 10.19861249810455011170905340845

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