↑

Properties

Label 2-48e2-48.35-c1-0-29
Degree $2$
Conductor $2304$
Sign $-0.955 + 0.296i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  + (−1.93 − 1.93i)5-s + 1.41·7-s + (−0.732 + 0.732i)11-s + (1.73 + 1.73i)13-s − 5.27i·17-s + (−5.27 + 5.27i)19-s − 3.46i·23-s + 2.46i·25-s + (2.31 − 2.31i)29-s − 9.14i·31-s + (−2.73 − 2.73i)35-s + (−2.46 + 2.46i)37-s − 4.52·41-s + (3.48 + 3.48i)43-s + 10.3·47-s + ⋯
L(s)  = 1  + (−0.863 − 0.863i)5-s + 0.534·7-s + (−0.220 + 0.220i)11-s + (0.480 + 0.480i)13-s − 1.28i·17-s + (−1.21 + 1.21i)19-s − 0.722i·23-s + 0.492i·25-s + (0.429 − 0.429i)29-s − 1.64i·31-s + (−0.461 − 0.461i)35-s + (−0.405 + 0.405i)37-s − 0.705·41-s + (0.531 + 0.531i)43-s + 1.51·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.955 + 0.296i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.955 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5822552777\)
\(L(\frac12)\) \(\approx\) \(0.5822552777\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (1.93 + 1.93i)T + 5iT^{2} \)
7 \( 1 - 1.41T + 7T^{2} \)
11 \( 1 + (0.732 - 0.732i)T - 11iT^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 + 5.27iT - 17T^{2} \)
19 \( 1 + (5.27 - 5.27i)T - 19iT^{2} \)
23 \( 1 + 3.46iT - 23T^{2} \)
29 \( 1 + (-2.31 + 2.31i)T - 29iT^{2} \)
31 \( 1 + 9.14iT - 31T^{2} \)
37 \( 1 + (2.46 - 2.46i)T - 37iT^{2} \)
41 \( 1 + 4.52T + 41T^{2} \)
43 \( 1 + (-3.48 - 3.48i)T + 43iT^{2} \)
47 \( 1 - 10.3T + 47T^{2} \)
53 \( 1 + (8.24 + 8.24i)T + 53iT^{2} \)
59 \( 1 + (8.92 - 8.92i)T - 59iT^{2} \)
61 \( 1 + (3 + 3i)T + 61iT^{2} \)
67 \( 1 + (-3.86 + 3.86i)T - 67iT^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 - 4.92iT - 73T^{2} \)
79 \( 1 - 2.17iT - 79T^{2} \)
83 \( 1 + (10.7 + 10.7i)T + 83iT^{2} \)
89 \( 1 + 16.1T + 89T^{2} \)
97 \( 1 + 18.3T + 97T^{2} \)
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

−8.338398257596632584950426070594, −8.198433181913145127990012714293, −7.25058541395348775118770763605, −6.31655974288669293251157928145, −5.38769759889771440137330750628, −4.34078791715341388242866824766, −4.19363369585290306310809722531, −2.71723357658490686711052761215, −1.53449666912448938945438866669, −0.20156230752001238148872033334, 1.52528528022211688623747116455, 2.83598452229769350730177130293, 3.59427615935889594969898649471, 4.44769523318494182633565188013, 5.41222223356595459832413983187, 6.39972764241737203753407315183, 7.04314968993918852487875712436, 7.87905434238711609187525055131, 8.456751761531373732094810151076, 9.149119419530770316048174878289

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