↑

Properties

Label 2-1412-1412.35-c0-0-0
Degree $2$
Conductor $1412$
Sign $0.855 - 0.517i$
Analytic cond. $0.704679$
Root an. cond. $0.839452$
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  + (0.959 + 0.281i)2-s + (0.841 + 0.540i)4-s + (−1.05 − 0.574i)5-s + (0.654 + 0.755i)8-s + (0.540 + 0.841i)9-s + (−0.847 − 0.847i)10-s + (0.334 − 0.898i)13-s + (0.415 + 0.909i)16-s + (1.10 + 1.27i)17-s + (0.281 + 0.959i)18-s + (−0.574 − 1.05i)20-s + (0.236 + 0.367i)25-s + (0.574 − 0.767i)26-s + (1.53 − 0.983i)29-s + (0.142 + 0.989i)32-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (0.841 + 0.540i)4-s + (−1.05 − 0.574i)5-s + (0.654 + 0.755i)8-s + (0.540 + 0.841i)9-s + (−0.847 − 0.847i)10-s + (0.334 − 0.898i)13-s + (0.415 + 0.909i)16-s + (1.10 + 1.27i)17-s + (0.281 + 0.959i)18-s + (−0.574 − 1.05i)20-s + (0.236 + 0.367i)25-s + (0.574 − 0.767i)26-s + (1.53 − 0.983i)29-s + (0.142 + 0.989i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1412 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1412\)    =    \(2^{2} \cdot 353\)
Sign: $0.855 - 0.517i$
Analytic conductor: \(0.704679\)
Root analytic conductor: \(0.839452\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1412} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1412,\ (\ :0),\ 0.855 - 0.517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.779645090\)
\(L(\frac12)\) \(\approx\) \(1.779645090\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 - 0.281i)T \)
353 \( 1 + (0.654 - 0.755i)T \)
good3 \( 1 + (-0.540 - 0.841i)T^{2} \)
5 \( 1 + (1.05 + 0.574i)T + (0.540 + 0.841i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.959 + 0.281i)T^{2} \)
13 \( 1 + (-0.334 + 0.898i)T + (-0.755 - 0.654i)T^{2} \)
17 \( 1 + (-1.10 - 1.27i)T + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (-0.654 - 0.755i)T^{2} \)
23 \( 1 + (-0.959 + 0.281i)T^{2} \)
29 \( 1 + (-1.53 + 0.983i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.281 + 0.959i)T^{2} \)
37 \( 1 + (1.98 + 0.142i)T + (0.989 + 0.142i)T^{2} \)
41 \( 1 + (1.29 + 0.186i)T + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (0.415 + 0.909i)T^{2} \)
53 \( 1 + (0.936 - 1.71i)T + (-0.540 - 0.841i)T^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (0.755 + 1.65i)T + (-0.654 + 0.755i)T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + (-0.909 + 0.415i)T^{2} \)
73 \( 1 + (0.415 + 0.909i)T^{2} \)
79 \( 1 + (-0.540 + 0.841i)T^{2} \)
83 \( 1 + (0.142 + 0.989i)T^{2} \)
89 \( 1 + (1.19 - 0.0855i)T + (0.989 - 0.142i)T^{2} \)
97 \( 1 + (-1.25 + 1.45i)T + (-0.142 - 0.989i)T^{2} \)
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

−10.19235750250620044067286526309, −8.481413763354961623047071994817, −8.112753927911177174027604368883, −7.49677606687615551228133103119, −6.46412513076586174880796957508, −5.48571843655142844723116268887, −4.75516346910468070963295282089, −3.92829837913539205015938369026, −3.14444829721069170770238982932, −1.62795174909827885768338659033, 1.38029605343241963760618271762, 3.04110145829307321021855032903, 3.54181059107609314500379930527, 4.47040082523471567557120240348, 5.31989951964427540038003847597, 6.69332730608349199224133140331, 6.87000889966478112508592115611, 7.79645722323226478367515712351, 8.954515584993972752755489578015, 9.918935570484859638128040841237

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