↑

Properties

Label 2-1472-1472.965-c0-0-0
Degree $2$
Conductor $1472$
Sign $0.956 + 0.290i$
Analytic cond. $0.734623$
Root an. cond. $0.857101$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.130 + 0.991i)2-s + (−1.65 − 1.10i)3-s + (−0.965 − 0.258i)4-s + (1.31 − 1.50i)6-s + (0.382 − 0.923i)8-s + (1.14 + 2.75i)9-s + (1.31 + 1.50i)12-s + (−0.867 + 0.172i)13-s + (0.866 + 0.5i)16-s + (−2.88 + 0.772i)18-s + (−0.923 + 0.382i)23-s + (−1.65 + 1.10i)24-s + (0.923 + 0.382i)25-s + (−0.0578 − 0.882i)26-s + (0.772 − 3.88i)27-s + ⋯
L(s)  = 1  + (−0.130 + 0.991i)2-s + (−1.65 − 1.10i)3-s + (−0.965 − 0.258i)4-s + (1.31 − 1.50i)6-s + (0.382 − 0.923i)8-s + (1.14 + 2.75i)9-s + (1.31 + 1.50i)12-s + (−0.867 + 0.172i)13-s + (0.866 + 0.5i)16-s + (−2.88 + 0.772i)18-s + (−0.923 + 0.382i)23-s + (−1.65 + 1.10i)24-s + (0.923 + 0.382i)25-s + (−0.0578 − 0.882i)26-s + (0.772 − 3.88i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.290i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $0.956 + 0.290i$
Analytic conductor: \(0.734623\)
Root analytic conductor: \(0.857101\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (965, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :0),\ 0.956 + 0.290i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4543829212\)
\(L(\frac12)\) \(\approx\) \(0.4543829212\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 - 0.991i)T \)
23 \( 1 + (0.923 - 0.382i)T \)
good3 \( 1 + (1.65 + 1.10i)T + (0.382 + 0.923i)T^{2} \)
5 \( 1 + (-0.923 - 0.382i)T^{2} \)
7 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (0.382 - 0.923i)T^{2} \)
13 \( 1 + (0.867 - 0.172i)T + (0.923 - 0.382i)T^{2} \)
17 \( 1 + iT^{2} \)
19 \( 1 + (0.923 - 0.382i)T^{2} \)
29 \( 1 + (-0.835 + 1.25i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + 0.261iT - T^{2} \)
37 \( 1 + (0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.923 + 0.382i)T + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.382 + 0.923i)T^{2} \)
47 \( 1 + (-1.36 + 1.36i)T - iT^{2} \)
53 \( 1 + (0.382 - 0.923i)T^{2} \)
59 \( 1 + (-1.63 - 0.324i)T + (0.923 + 0.382i)T^{2} \)
61 \( 1 + (-0.382 - 0.923i)T^{2} \)
67 \( 1 + (-0.382 - 0.923i)T^{2} \)
71 \( 1 + (-0.465 + 1.12i)T + (-0.707 - 0.707i)T^{2} \)
73 \( 1 + (0.198 + 0.478i)T + (-0.707 + 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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

−9.811772040876310979630652854476, −8.521398670136778805968654425278, −7.65950847409518756285824197742, −7.15075675632878625126705799754, −6.42713481063632636894959209169, −5.73138629060358143600060927331, −5.05782910978178391219082631047, −4.23820124001746296333588215319, −2.10730566676595812993243881948, −0.65235050718403857882520871094, 0.978783891593516166102316244008, 2.81704852263782768181217729860, 3.97380501435881725004539130005, 4.66669424761420122470079992938, 5.28147314242715698096527814604, 6.19246428182255158183765557650, 7.18559625719388141021227405883, 8.505959209877949006634363072365, 9.416305322388503758347332641281, 9.951497916503560686363752477532

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