↑

Properties

Label 2-714-17.4-c1-0-11
Degree $2$
Conductor $714$
Sign $-0.967 + 0.254i$
Analytic cond. $5.70131$
Root an. cond. $2.38774$
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  − i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−2.39 + 2.39i)5-s + (0.707 + 0.707i)6-s + (0.707 + 0.707i)7-s + i·8-s − 1.00i·9-s + (2.39 + 2.39i)10-s + (1.39 + 1.39i)11-s + (0.707 − 0.707i)12-s − 1.28·13-s + (0.707 − 0.707i)14-s − 3.38i·15-s + 16-s + (−3.78 − 1.62i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−1.06 + 1.06i)5-s + (0.288 + 0.288i)6-s + (0.267 + 0.267i)7-s + 0.353i·8-s − 0.333i·9-s + (0.755 + 0.755i)10-s + (0.419 + 0.419i)11-s + (0.204 − 0.204i)12-s − 0.356·13-s + (0.188 − 0.188i)14-s − 0.872i·15-s + 0.250·16-s + (−0.918 − 0.394i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(714\)    =    \(2 \cdot 3 \cdot 7 \cdot 17\)
Sign: $-0.967 + 0.254i$
Analytic conductor: \(5.70131\)
Root analytic conductor: \(2.38774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{714} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 714,\ (\ :1/2),\ -0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0125305 - 0.0967449i\)
\(L(\frac12)\) \(\approx\) \(0.0125305 - 0.0967449i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (0.707 - 0.707i)T \)
7 \( 1 + (-0.707 - 0.707i)T \)
17 \( 1 + (3.78 + 1.62i)T \)
good5 \( 1 + (2.39 - 2.39i)T - 5iT^{2} \)
11 \( 1 + (-1.39 - 1.39i)T + 11iT^{2} \)
13 \( 1 + 1.28T + 13T^{2} \)
19 \( 1 + 8.42iT - 19T^{2} \)
23 \( 1 + (2.44 + 2.44i)T + 23iT^{2} \)
29 \( 1 + (2.63 - 2.63i)T - 29iT^{2} \)
31 \( 1 + (4.74 - 4.74i)T - 31iT^{2} \)
37 \( 1 + (1.22 - 1.22i)T - 37iT^{2} \)
41 \( 1 + (-1.33 - 1.33i)T + 41iT^{2} \)
43 \( 1 + 12.0iT - 43T^{2} \)
47 \( 1 + 5.10T + 47T^{2} \)
53 \( 1 + 13.8iT - 53T^{2} \)
59 \( 1 - 12.5iT - 59T^{2} \)
61 \( 1 + (7.70 + 7.70i)T + 61iT^{2} \)
67 \( 1 - 15.9T + 67T^{2} \)
71 \( 1 + (5.22 - 5.22i)T - 71iT^{2} \)
73 \( 1 + (2.40 - 2.40i)T - 73iT^{2} \)
79 \( 1 + (9.16 + 9.16i)T + 79iT^{2} \)
83 \( 1 - 7.05iT - 83T^{2} \)
89 \( 1 - 0.0821T + 89T^{2} \)
97 \( 1 + (-2.30 + 2.30i)T - 97iT^{2} \)
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

−10.31210714731804683270908070842, −9.273843273490580196998368390436, −8.531132685379083758020287779875, −7.19728321379117415146657116485, −6.74951326972270108766638696585, −5.16572235240441216053755566786, −4.35567450661841498822965949343, −3.38741581646564997974361425306, −2.30800487648943450304917289395, −0.05501469059000513375901237797, 1.49254243336529740096824176941, 3.81502717725192662380546460967, 4.43344252134554246218983048970, 5.56018997109078084478210712927, 6.34104435570670641385448089126, 7.67487546218724807811442319940, 7.902511053574378187472084960624, 8.829749195750772983562167832118, 9.765739037404426767186358797561, 11.01580430872302327591147346616

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