↑

Properties

Label 2-3024-63.59-c1-0-2
Degree $2$
Conductor $3024$
Sign $-0.874 + 0.484i$
Analytic cond. $24.1467$
Root an. cond. $4.91393$
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  − 2.04·5-s + (−1.41 + 2.23i)7-s + 5.90i·11-s + (−0.139 − 0.0804i)13-s + (−2.77 + 4.81i)17-s + (4.02 − 2.32i)19-s + 0.433i·23-s − 0.801·25-s + (1.95 − 1.12i)29-s + (2.57 − 1.48i)31-s + (2.89 − 4.58i)35-s + (2.17 + 3.77i)37-s + (−2.35 + 4.08i)41-s + (−1.82 − 3.15i)43-s + (0.0650 − 0.112i)47-s + ⋯
L(s)  = 1  − 0.916·5-s + (−0.534 + 0.845i)7-s + 1.78i·11-s + (−0.0386 − 0.0223i)13-s + (−0.674 + 1.16i)17-s + (0.922 − 0.532i)19-s + 0.0903i·23-s − 0.160·25-s + (0.363 − 0.209i)29-s + (0.463 − 0.267i)31-s + (0.489 − 0.774i)35-s + (0.358 + 0.620i)37-s + (−0.368 + 0.637i)41-s + (−0.278 − 0.481i)43-s + (0.00948 − 0.0164i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.874 + 0.484i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3024\)    =    \(2^{4} \cdot 3^{3} \cdot 7\)
Sign: $-0.874 + 0.484i$
Analytic conductor: \(24.1467\)
Root analytic conductor: \(4.91393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3024} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3024,\ (\ :1/2),\ -0.874 + 0.484i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3301455538\)
\(L(\frac12)\) \(\approx\) \(0.3301455538\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 + (1.41 - 2.23i)T \)
good5 \( 1 + 2.04T + 5T^{2} \)
11 \( 1 - 5.90iT - 11T^{2} \)
13 \( 1 + (0.139 + 0.0804i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.77 - 4.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.02 + 2.32i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 0.433iT - 23T^{2} \)
29 \( 1 + (-1.95 + 1.12i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.57 + 1.48i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.17 - 3.77i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.35 - 4.08i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.82 + 3.15i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.0650 + 0.112i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (10.7 + 6.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.22 + 5.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.98 + 3.45i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.64 - 13.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 1.48iT - 71T^{2} \)
73 \( 1 + (2.60 + 1.50i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-8.69 + 15.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.62 - 13.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.04 + 7.01i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.61 + 1.50i)T + (48.5 - 84.0i)T^{2} \)
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

−9.259638120821446066202242101738, −8.257963456609518079294724227622, −7.77849943823939716940984430276, −6.84217081464451901323375607944, −6.33224546698118303485627073487, −5.17223199289564201058274225737, −4.50900651427441554191838647386, −3.67010620262974036457293333711, −2.67468263064045766079972365337, −1.70703695169408807551051049638, 0.11999935210874710565082254728, 1.04626255264327135585073831565, 2.86645974336631672890526039310, 3.44470978198863629591702579745, 4.20945403454582212653406621985, 5.15434409279670058194381209852, 6.11957930053350232260605796883, 6.81202694365233558286556420270, 7.67048755990565875368183925772, 8.100523621720390573916036139601

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