Properties

Label 2-3800-5.4-c1-0-62
Degree $2$
Conductor $3800$
Sign $0.894 + 0.447i$
Analytic cond. $30.3431$
Root an. cond. $5.50846$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.08i·3-s + 4.29i·7-s − 6.51·9-s − 1.21·11-s − 5.08i·13-s − 2.29i·17-s + 19-s − 13.2·21-s − 7.67i·23-s − 10.8i·27-s + 0.489·29-s − 3.74i·33-s − 2i·37-s + 15.6·39-s − 4.16·41-s + ⋯
L(s)  = 1  + 1.78i·3-s + 1.62i·7-s − 2.17·9-s − 0.365·11-s − 1.41i·13-s − 0.557i·17-s + 0.229·19-s − 2.89·21-s − 1.60i·23-s − 2.08i·27-s + 0.0909·29-s − 0.651i·33-s − 0.328i·37-s + 2.51·39-s − 0.650·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.894 + 0.447i$
Analytic conductor: \(30.3431\)
Root analytic conductor: \(5.50846\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :1/2),\ 0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6123491332\)
\(L(\frac12)\) \(\approx\) \(0.6123491332\)
\(L(\frac{3}{2})\) 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 \( 1 \)
19 \( 1 - T \)
good3 \( 1 - 3.08iT - 3T^{2} \)
7 \( 1 - 4.29iT - 7T^{2} \)
11 \( 1 + 1.21T + 11T^{2} \)
13 \( 1 + 5.08iT - 13T^{2} \)
17 \( 1 + 2.29iT - 17T^{2} \)
23 \( 1 + 7.67iT - 23T^{2} \)
29 \( 1 - 0.489T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 4.16T + 41T^{2} \)
43 \( 1 + 12.9iT - 43T^{2} \)
47 \( 1 + 5.80iT - 47T^{2} \)
53 \( 1 + 1.93iT - 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 5.38T + 61T^{2} \)
67 \( 1 - 2.48iT - 67T^{2} \)
71 \( 1 + 11.7T + 71T^{2} \)
73 \( 1 - 8.46iT - 73T^{2} \)
79 \( 1 + 1.83T + 79T^{2} \)
83 \( 1 - 7.02iT - 83T^{2} \)
89 \( 1 + 13.7T + 89T^{2} \)
97 \( 1 + 3.57iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.590870865685355767489217060915, −8.179836837340462387887825771707, −6.86659583918618434549370964103, −5.61204123343592595059291395406, −5.52510651675324663879447766850, −4.79207056184273708729411307530, −3.81834009142702967632493230365, −2.89920171658089562192851956218, −2.45630065209201146777977592173, −0.18117613672399972289656012796, 1.23088984517142501884850294330, 1.64346976005425093249839407508, 2.89891829240674223251911496352, 3.85676706846165223567941759290, 4.76911716785583111884395203470, 5.91339900931436587634682368140, 6.56008320259746766802898049846, 7.16627588304168852570438866281, 7.63160751573488777814412398505, 8.189768577543337456022691851440

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