↑

Properties

Label 2-3332-3332.1835-c0-0-1
Degree $2$
Conductor $3332$
Sign $0.926 - 0.375i$
Analytic cond. $1.66288$
Root an. cond. $1.28952$
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  + (−0.900 + 0.433i)2-s + (−0.0990 + 0.433i)3-s + (0.623 − 0.781i)4-s + (−0.0990 − 0.433i)6-s + (0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.722 + 0.347i)9-s + (−1.62 + 0.781i)11-s + (0.277 + 0.347i)12-s + (1.62 − 0.781i)13-s + (0.222 + 0.974i)14-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 0.801·18-s + (0.400 + 0.193i)21-s + (1.12 − 1.40i)22-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−0.0990 + 0.433i)3-s + (0.623 − 0.781i)4-s + (−0.0990 − 0.433i)6-s + (0.222 − 0.974i)7-s + (−0.222 + 0.974i)8-s + (0.722 + 0.347i)9-s + (−1.62 + 0.781i)11-s + (0.277 + 0.347i)12-s + (1.62 − 0.781i)13-s + (0.222 + 0.974i)14-s + (−0.222 − 0.974i)16-s + (0.623 + 0.781i)17-s − 0.801·18-s + (0.400 + 0.193i)21-s + (1.12 − 1.40i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.926 - 0.375i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3332\)    =    \(2^{2} \cdot 7^{2} \cdot 17\)
Sign: $0.926 - 0.375i$
Analytic conductor: \(1.66288\)
Root analytic conductor: \(1.28952\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3332} (1835, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3332,\ (\ :0),\ 0.926 - 0.375i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8832451308\)
\(L(\frac12)\) \(\approx\) \(0.8832451308\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.900 - 0.433i)T \)
7 \( 1 + (-0.222 + 0.974i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
good3 \( 1 + (0.0990 - 0.433i)T + (-0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.900 + 0.433i)T^{2} \)
11 \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \)
13 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.12 + 1.40i)T + (-0.222 - 0.974i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 - 0.445T + T^{2} \)
37 \( 1 + (0.222 - 0.974i)T^{2} \)
41 \( 1 + (0.900 + 0.433i)T^{2} \)
43 \( 1 + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
59 \( 1 + (0.900 - 0.433i)T^{2} \)
61 \( 1 + (0.222 - 0.974i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.277 + 0.347i)T + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-0.623 - 0.781i)T^{2} \)
79 \( 1 - 1.80T + T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
97 \( 1 - T^{2} \)
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

−8.657564751605842012991432171992, −7.991147807054068463765529260028, −7.64598103679234155626521375761, −6.76973175979126232891423717370, −5.94375046224691988838376135524, −5.08604037242872452851460728849, −4.40656388852732673871962817782, −3.29066087686150034731770777356, −2.05500255540446985087323566083, −0.936295276039083458896097147773, 1.06748305101418593736976033204, 1.98518704187982317773688742129, 3.03205587319924430838075138902, 3.70971037076037997130686393964, 5.11479966696119749304116492927, 5.89527135838287467227718524356, 6.63050906769722321567733556698, 7.60080360299507606931143394431, 7.981246875049074950077669674280, 8.852515983537283689492893576032

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