↑

Properties

Label 2-1368-152.75-c1-0-26
Degree $2$
Conductor $1368$
Sign $-0.820 - 0.571i$
Analytic cond. $10.9235$
Root an. cond. $3.30507$
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  + (−1.39 + 0.244i)2-s + (1.88 − 0.680i)4-s + 3.10i·5-s + 4.34i·7-s + (−2.45 + 1.40i)8-s + (−0.757 − 4.32i)10-s − 2.65·11-s + 5.10·13-s + (−1.06 − 6.05i)14-s + (3.07 − 2.55i)16-s + 3.48·17-s + (1.86 + 3.93i)19-s + (2.11 + 5.83i)20-s + (3.70 − 0.648i)22-s + 5.31i·23-s + ⋯
L(s)  = 1  + (−0.984 + 0.172i)2-s + (0.940 − 0.340i)4-s + 1.38i·5-s + 1.64i·7-s + (−0.867 + 0.497i)8-s + (−0.239 − 1.36i)10-s − 0.800·11-s + 1.41·13-s + (−0.283 − 1.61i)14-s + (0.768 − 0.639i)16-s + 0.846·17-s + (0.427 + 0.903i)19-s + (0.471 + 1.30i)20-s + (0.788 − 0.138i)22-s + 1.10i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 - 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1368\)    =    \(2^{3} \cdot 3^{2} \cdot 19\)
Sign: $-0.820 - 0.571i$
Analytic conductor: \(10.9235\)
Root analytic conductor: \(3.30507\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1368} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1368,\ (\ :1/2),\ -0.820 - 0.571i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.068294226\)
\(L(\frac12)\) \(\approx\) \(1.068294226\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.244i)T \)
3 \( 1 \)
19 \( 1 + (-1.86 - 3.93i)T \)
good5 \( 1 - 3.10iT - 5T^{2} \)
7 \( 1 - 4.34iT - 7T^{2} \)
11 \( 1 + 2.65T + 11T^{2} \)
13 \( 1 - 5.10T + 13T^{2} \)
17 \( 1 - 3.48T + 17T^{2} \)
23 \( 1 - 5.31iT - 23T^{2} \)
29 \( 1 - 7.98T + 29T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + 4.57T + 37T^{2} \)
41 \( 1 + 4.51iT - 41T^{2} \)
43 \( 1 - 6.28T + 43T^{2} \)
47 \( 1 - 7.68iT - 47T^{2} \)
53 \( 1 + 9.81T + 53T^{2} \)
59 \( 1 + 4.94iT - 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + 0.406iT - 67T^{2} \)
71 \( 1 + 2.40T + 71T^{2} \)
73 \( 1 - 0.373T + 73T^{2} \)
79 \( 1 + 17.2T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 1.95iT - 89T^{2} \)
97 \( 1 - 4.87iT - 97T^{2} \)
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

−9.897934334484569432129709293325, −9.118028651017323710813309969998, −8.197300381590959721911164313625, −7.75658141842228980564510412546, −6.60723982487018952869848675788, −5.98736902033547593240797606815, −5.39469002288153597828815201196, −3.31164529755196667281134910538, −2.80502308097295143316395437694, −1.63593530071624880104370463079, 0.71069817476255479293430227921, 1.22885352592693599262976797406, 2.95617288314445758282354573284, 4.07659418622466373204639124967, 4.94838713406220646495816335700, 6.15144897767806598853183469635, 7.05768084245718100539404208440, 7.918876817818175332456706348576, 8.458784966691323458579939054680, 9.138262042658521409063466613632

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