↑

Properties

Label 2-4100-5.4-c1-0-39
Degree $2$
Conductor $4100$
Sign $0.894 + 0.447i$
Analytic cond. $32.7386$
Root an. cond. $5.72177$
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  + 1.68i·3-s − 4.03i·7-s + 0.149·9-s + 0.693·11-s + 2.78i·13-s − 0.285i·17-s − 2.88·19-s + 6.81·21-s − 6.88i·23-s + 5.31i·27-s − 3.46·29-s + 8.71·31-s + 1.17i·33-s + 1.40i·37-s − 4.69·39-s + ⋯
L(s)  = 1  + 0.974i·3-s − 1.52i·7-s + 0.0499·9-s + 0.208·11-s + 0.771i·13-s − 0.0692i·17-s − 0.662·19-s + 1.48·21-s − 1.43i·23-s + 1.02i·27-s − 0.643·29-s + 1.56·31-s + 0.203i·33-s + 0.231i·37-s − 0.752·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4100\)    =    \(2^{2} \cdot 5^{2} \cdot 41\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(32.7386\)
Root analytic conductor: \(5.72177\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4100} (1149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4100,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.782993446\)
\(L(\frac12)\) \(\approx\) \(1.782993446\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
41 \( 1 - T \)
good3 \( 1 - 1.68iT - 3T^{2} \)
7 \( 1 + 4.03iT - 7T^{2} \)
11 \( 1 - 0.693T + 11T^{2} \)
13 \( 1 - 2.78iT - 13T^{2} \)
17 \( 1 + 0.285iT - 17T^{2} \)
19 \( 1 + 2.88T + 19T^{2} \)
23 \( 1 + 6.88iT - 23T^{2} \)
29 \( 1 + 3.46T + 29T^{2} \)
31 \( 1 - 8.71T + 31T^{2} \)
37 \( 1 - 1.40iT - 37T^{2} \)
43 \( 1 - 0.924iT - 43T^{2} \)
47 \( 1 + 8.52iT - 47T^{2} \)
53 \( 1 - 3.50iT - 53T^{2} \)
59 \( 1 - 4.69T + 59T^{2} \)
61 \( 1 + 0.199T + 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 - 1.95T + 71T^{2} \)
73 \( 1 + 5.03iT - 73T^{2} \)
79 \( 1 - 5.73T + 79T^{2} \)
83 \( 1 + 5.03iT - 83T^{2} \)
89 \( 1 - 10.7T + 89T^{2} \)
97 \( 1 + 13.6iT - 97T^{2} \)
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

−8.453285750761369141274816922570, −7.62650304634460796378634066128, −6.80386212123527978370281116573, −6.39060059037796953963976133257, −5.08626642611304544448039725741, −4.29023694449963076186114727601, −4.14788919088776791830358809757, −3.13612913878102407384582259998, −1.83888084631261833907895768548, −0.58263077516067245684375935101, 1.05575164857758221879092967626, 2.07415072082393405392686965240, 2.72292539196006770510829037036, 3.78895505924072643498023816494, 4.92181892205174932591010954683, 5.72298867264461631580837578906, 6.22114038578705986525572099171, 7.00881196311376637696038769569, 7.86610381377797925536290372879, 8.283651702436526882270482406704

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