Properties

Label 2-1152-8.3-c2-0-31
Degree $2$
Conductor $1152$
Sign $i$
Analytic cond. $31.3897$
Root an. cond. $5.60265$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2i·5-s − 9.79i·7-s + 19.5·11-s + 21·25-s − 50i·29-s + 48.9i·31-s − 19.5·35-s − 46.9·49-s − 94i·53-s − 39.1i·55-s − 117.·59-s + 50·73-s − 191. i·77-s − 146. i·79-s + 97.9·83-s + ⋯
L(s)  = 1  − 0.400i·5-s − 1.39i·7-s + 1.78·11-s + 0.839·25-s − 1.72i·29-s + 1.58i·31-s − 0.559·35-s − 0.959·49-s − 1.77i·53-s − 0.712i·55-s − 1.99·59-s + 0.684·73-s − 2.49i·77-s − 1.86i·79-s + 1.18·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1152\)    =    \(2^{7} \cdot 3^{2}\)
Sign: $i$
Analytic conductor: \(31.3897\)
Root analytic conductor: \(5.60265\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1152} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1152,\ (\ :1),\ i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.034937300\)
\(L(\frac12)\) \(\approx\) \(2.034937300\)
\(L(2)\) 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 \( 1 \)
good5 \( 1 + 2iT - 25T^{2} \)
7 \( 1 + 9.79iT - 49T^{2} \)
11 \( 1 - 19.5T + 121T^{2} \)
13 \( 1 - 169T^{2} \)
17 \( 1 + 289T^{2} \)
19 \( 1 + 361T^{2} \)
23 \( 1 - 529T^{2} \)
29 \( 1 + 50iT - 841T^{2} \)
31 \( 1 - 48.9iT - 961T^{2} \)
37 \( 1 - 1.36e3T^{2} \)
41 \( 1 + 1.68e3T^{2} \)
43 \( 1 + 1.84e3T^{2} \)
47 \( 1 - 2.20e3T^{2} \)
53 \( 1 + 94iT - 2.80e3T^{2} \)
59 \( 1 + 117.T + 3.48e3T^{2} \)
61 \( 1 - 3.72e3T^{2} \)
67 \( 1 + 4.48e3T^{2} \)
71 \( 1 - 5.04e3T^{2} \)
73 \( 1 - 50T + 5.32e3T^{2} \)
79 \( 1 + 146. iT - 6.24e3T^{2} \)
83 \( 1 - 97.9T + 6.88e3T^{2} \)
89 \( 1 + 7.92e3T^{2} \)
97 \( 1 + 190T + 9.40e3T^{2} \)
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

−9.398830896493685616318733553266, −8.660070369244321336374511831787, −7.70562536347938761484195003161, −6.83483314747049142690532555878, −6.27504876914897286755346824105, −4.87615764422648190118350236222, −4.14826242169324120249194317948, −3.35846041890088462009654509504, −1.61123040821872616575888505994, −0.68079489459746825856764101310, 1.35677478127816702379097429599, 2.54688546269181576070764416388, 3.54006879705857753769415358668, 4.64071195967178653338179789815, 5.74194017524548546685894363839, 6.40256164978630995380455819596, 7.19811123482758943996720094233, 8.357355243561090251645568465293, 9.146015386788151476522327629068, 9.466309394877752989834197825896

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