↑

Properties

Label 2-920-115.7-c1-0-29
Degree $2$
Conductor $920$
Sign $0.402 + 0.915i$
Analytic cond. $7.34623$
Root an. cond. $2.71039$
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  + (−0.664 + 3.05i)3-s + (0.777 − 2.09i)5-s + (0.825 − 1.51i)7-s + (−6.17 − 2.81i)9-s + (1.45 − 1.25i)11-s + (−2.76 + 1.50i)13-s + (5.89 + 3.77i)15-s + (−3.78 − 5.05i)17-s + (−0.770 − 5.36i)19-s + (4.07 + 3.52i)21-s + (−2.69 − 3.96i)23-s + (−3.79 − 3.26i)25-s + (7.09 − 9.48i)27-s + (−7.28 − 1.04i)29-s + (−4.63 − 2.98i)31-s + ⋯
L(s)  = 1  + (−0.383 + 1.76i)3-s + (0.347 − 0.937i)5-s + (0.312 − 0.571i)7-s + (−2.05 − 0.939i)9-s + (0.437 − 0.379i)11-s + (−0.765 + 0.418i)13-s + (1.52 + 0.973i)15-s + (−0.917 − 1.22i)17-s + (−0.176 − 1.23i)19-s + (0.888 + 0.770i)21-s + (−0.562 − 0.826i)23-s + (−0.758 − 0.652i)25-s + (1.36 − 1.82i)27-s + (−1.35 − 0.194i)29-s + (−0.833 − 0.535i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 + 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(920\)    =    \(2^{3} \cdot 5 \cdot 23\)
Sign: $0.402 + 0.915i$
Analytic conductor: \(7.34623\)
Root analytic conductor: \(2.71039\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{920} (697, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 920,\ (\ :1/2),\ 0.402 + 0.915i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.688238 - 0.449193i\)
\(L(\frac12)\) \(\approx\) \(0.688238 - 0.449193i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-0.777 + 2.09i)T \)
23 \( 1 + (2.69 + 3.96i)T \)
good3 \( 1 + (0.664 - 3.05i)T + (-2.72 - 1.24i)T^{2} \)
7 \( 1 + (-0.825 + 1.51i)T + (-3.78 - 5.88i)T^{2} \)
11 \( 1 + (-1.45 + 1.25i)T + (1.56 - 10.8i)T^{2} \)
13 \( 1 + (2.76 - 1.50i)T + (7.02 - 10.9i)T^{2} \)
17 \( 1 + (3.78 + 5.05i)T + (-4.78 + 16.3i)T^{2} \)
19 \( 1 + (0.770 + 5.36i)T + (-18.2 + 5.35i)T^{2} \)
29 \( 1 + (7.28 + 1.04i)T + (27.8 + 8.17i)T^{2} \)
31 \( 1 + (4.63 + 2.98i)T + (12.8 + 28.1i)T^{2} \)
37 \( 1 + (-2.41 - 6.47i)T + (-27.9 + 24.2i)T^{2} \)
41 \( 1 + (2.27 + 4.98i)T + (-26.8 + 30.9i)T^{2} \)
43 \( 1 + (-5.80 - 1.26i)T + (39.1 + 17.8i)T^{2} \)
47 \( 1 + (-9.54 - 9.54i)T + 47iT^{2} \)
53 \( 1 + (4.03 + 2.20i)T + (28.6 + 44.5i)T^{2} \)
59 \( 1 + (1.46 - 4.97i)T + (-49.6 - 31.8i)T^{2} \)
61 \( 1 + (-6.07 + 9.45i)T + (-25.3 - 55.4i)T^{2} \)
67 \( 1 + (-0.747 - 10.4i)T + (-66.3 + 9.53i)T^{2} \)
71 \( 1 + (-0.656 + 0.757i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (2.68 + 2.00i)T + (20.5 + 70.0i)T^{2} \)
79 \( 1 + (6.98 + 2.05i)T + (66.4 + 42.7i)T^{2} \)
83 \( 1 + (-13.4 + 5.02i)T + (62.7 - 54.3i)T^{2} \)
89 \( 1 + (2.34 - 1.50i)T + (36.9 - 80.9i)T^{2} \)
97 \( 1 + (-0.484 - 0.180i)T + (73.3 + 63.5i)T^{2} \)
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

−9.732889320530048065334078391789, −9.253598338459580536519849151790, −8.760911951750340774542160932734, −7.42118944568100523279114608587, −6.19347120631391243041485219806, −5.21824743089954480820424755721, −4.52981936113303199699589577217, −4.06295887623906316194264564539, −2.51929209834378481501455507280, −0.38179702769717493171006191946, 1.80710295166208431790830685196, 2.20201315734094757169762423467, 3.72279562736975721142573330881, 5.58770637826388289179489369234, 5.92912549115568863316329603098, 6.93523268484259644848618539269, 7.48569723793140521978143174131, 8.265879333935875967474734638637, 9.310455432213672679283851206781, 10.47224750421446443151759806538

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