↑

Properties

Label 2-1008-63.47-c1-0-27
Degree $2$
Conductor $1008$
Sign $0.994 + 0.107i$
Analytic cond. $8.04892$
Root an. cond. $2.83706$
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  + (1.70 − 0.278i)3-s − 0.542·5-s + (2.62 + 0.340i)7-s + (2.84 − 0.950i)9-s + 0.769i·11-s + (2.96 − 1.71i)13-s + (−0.926 + 0.150i)15-s + (3.23 + 5.60i)17-s + (−5.60 − 3.23i)19-s + (4.58 − 0.148i)21-s + 0.115i·23-s − 4.70·25-s + (4.59 − 2.41i)27-s + (4.40 + 2.54i)29-s + (−4.01 − 2.31i)31-s + ⋯
L(s)  = 1  + (0.987 − 0.160i)3-s − 0.242·5-s + (0.991 + 0.128i)7-s + (0.948 − 0.316i)9-s + 0.231i·11-s + (0.822 − 0.474i)13-s + (−0.239 + 0.0389i)15-s + (0.784 + 1.35i)17-s + (−1.28 − 0.742i)19-s + (0.999 − 0.0323i)21-s + 0.0241i·23-s − 0.941·25-s + (0.885 − 0.465i)27-s + (0.817 + 0.471i)29-s + (−0.720 − 0.416i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1008\)    =    \(2^{4} \cdot 3^{2} \cdot 7\)
Sign: $0.994 + 0.107i$
Analytic conductor: \(8.04892\)
Root analytic conductor: \(2.83706\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1008} (929, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1008,\ (\ :1/2),\ 0.994 + 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.520098993\)
\(L(\frac12)\) \(\approx\) \(2.520098993\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.278i)T \)
7 \( 1 + (-2.62 - 0.340i)T \)
good5 \( 1 + 0.542T + 5T^{2} \)
11 \( 1 - 0.769iT - 11T^{2} \)
13 \( 1 + (-2.96 + 1.71i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.23 - 5.60i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.60 + 3.23i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 0.115iT - 23T^{2} \)
29 \( 1 + (-4.40 - 2.54i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.01 + 2.31i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.47 + 9.48i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.04 - 7.00i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (3.32 - 5.76i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.773 + 1.33i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.221 + 0.127i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.12 + 8.86i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.83 + 2.78i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.64 - 2.84i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 5.67iT - 71T^{2} \)
73 \( 1 + (5.35 - 3.09i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2.01 - 3.48i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.80 - 10.0i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.00 + 3.47i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.0 + 8.69i)T + (48.5 + 84.0i)T^{2} \)
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

−9.882635754333012795002198830239, −8.908751951543950301817324929555, −8.141261851466320786054426690961, −7.86331269692668906365022754236, −6.67050179139425926632169732459, −5.68216793702775804309316496202, −4.39468538797874374025388087561, −3.73393991539659155600660449832, −2.42728852970045619719188280023, −1.38433417068587061127162615496, 1.38667266360701501715303977154, 2.55454298129836298873664205033, 3.78754458875614377750390845786, 4.44951656558087679907373137183, 5.57504165053865953971616729449, 6.80754465827612420898118170220, 7.72240818449112096733216154493, 8.319932718391859035846841811410, 8.965288566970819657743548226485, 9.954524957632185747999493774357

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