↑

Properties

Label 2-644-644.419-c0-0-1
Degree $2$
Conductor $644$
Sign $0.555 + 0.831i$
Analytic cond. $0.321397$
Root an. cond. $0.566919$
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  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.415 − 0.909i)7-s + (0.841 − 0.540i)8-s + (0.142 + 0.989i)9-s + (1.25 − 0.368i)11-s + (−0.415 + 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)18-s + (−1.10 − 0.708i)22-s + (0.142 − 0.989i)23-s + (0.959 + 0.281i)25-s + (0.959 − 0.281i)28-s + (1.61 − 1.03i)29-s + (0.415 + 0.909i)32-s + ⋯
L(s)  = 1  + (−0.654 − 0.755i)2-s + (−0.142 + 0.989i)4-s + (−0.415 − 0.909i)7-s + (0.841 − 0.540i)8-s + (0.142 + 0.989i)9-s + (1.25 − 0.368i)11-s + (−0.415 + 0.909i)14-s + (−0.959 − 0.281i)16-s + (0.654 − 0.755i)18-s + (−1.10 − 0.708i)22-s + (0.142 − 0.989i)23-s + (0.959 + 0.281i)25-s + (0.959 − 0.281i)28-s + (1.61 − 1.03i)29-s + (0.415 + 0.909i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 644 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(644\)    =    \(2^{2} \cdot 7 \cdot 23\)
Sign: $0.555 + 0.831i$
Analytic conductor: \(0.321397\)
Root analytic conductor: \(0.566919\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{644} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 644,\ (\ :0),\ 0.555 + 0.831i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6942517161\)
\(L(\frac12)\) \(\approx\) \(0.6942517161\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.654 + 0.755i)T \)
7 \( 1 + (0.415 + 0.909i)T \)
23 \( 1 + (-0.142 + 0.989i)T \)
good3 \( 1 + (-0.142 - 0.989i)T^{2} \)
5 \( 1 + (-0.959 - 0.281i)T^{2} \)
11 \( 1 + (-1.25 + 0.368i)T + (0.841 - 0.540i)T^{2} \)
13 \( 1 + (0.654 + 0.755i)T^{2} \)
17 \( 1 + (0.415 - 0.909i)T^{2} \)
19 \( 1 + (-0.415 - 0.909i)T^{2} \)
29 \( 1 + (-1.61 + 1.03i)T + (0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + (1.80 - 0.258i)T + (0.959 - 0.281i)T^{2} \)
41 \( 1 + (0.959 + 0.281i)T^{2} \)
43 \( 1 + (0.186 - 0.215i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.512 + 0.234i)T + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.654 - 0.755i)T^{2} \)
61 \( 1 + (-0.142 + 0.989i)T^{2} \)
67 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
71 \( 1 + (0.425 - 1.45i)T + (-0.841 - 0.540i)T^{2} \)
73 \( 1 + (-0.415 - 0.909i)T^{2} \)
79 \( 1 + (0.797 - 1.74i)T + (-0.654 - 0.755i)T^{2} \)
83 \( 1 + (0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.959 - 0.281i)T^{2} \)
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

−10.47817800310418081126840106927, −10.05242419495956548654435936498, −8.913093686202298806874220975489, −8.273384462600119347047139205131, −7.18109364642807956611076692087, −6.51719355649798010057805395350, −4.76492283709239437785051430701, −3.91020713519631116002393765543, −2.76059965069675619935608786794, −1.24186542486451912069153592232, 1.48394625886130511999520856478, 3.26460258384793360858945680996, 4.66080793911582775544133417558, 5.78449099196766453900676907802, 6.61832947013023238637297104189, 7.14870457669270003037284207510, 8.673892775694993491609128997623, 8.962730876531362565342530670798, 9.760865191231838588358588754973, 10.63193944748768409882484324887

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