Properties

Label 2-3800-5.4-c1-0-39
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  + i·3-s − 3i·7-s + 2·9-s + 2·11-s + i·13-s + 5i·17-s − 19-s + 3·21-s i·23-s + 5i·27-s + 3·29-s + 4·31-s + 2i·33-s − 2i·37-s − 39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.13i·7-s + 0.666·9-s + 0.603·11-s + 0.277i·13-s + 1.21i·17-s − 0.229·19-s + 0.654·21-s − 0.208i·23-s + 0.962i·27-s + 0.557·29-s + 0.718·31-s + 0.348i·33-s − 0.328i·37-s − 0.160·39-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

\(L(1)\) \(\approx\) \(2.131214705\)
\(L(\frac12)\) \(\approx\) \(2.131214705\)
\(L(\frac{3}{2})\) 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 \( 1 - iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 - 5iT - 17T^{2} \)
23 \( 1 + iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 - 4T + 31T^{2} \)
37 \( 1 + 2iT - 37T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 8iT - 43T^{2} \)
47 \( 1 - 8iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + T + 59T^{2} \)
61 \( 1 - 14T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 - 10T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 10iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + 14iT - 97T^{2} \)
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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

−8.528025274072942944335659489349, −7.87812407280192587973044173695, −6.88750124009191480934460013894, −6.59521731372866378725017474374, −5.48266750040847014794515410383, −4.44301408788742981283387886075, −4.09453531427954223741771908931, −3.36063433232076111183966528523, −1.92830454358131128596931199848, −0.943079307186390359816593907397, 0.824261714270010477597282677940, 1.94866917095304004562577190739, 2.72860112158830787645053082073, 3.74311036894514553072594356463, 4.82924182779664761793597756632, 5.39679155135859262068134042788, 6.53625648618392145621412596375, 6.72798501579055834463631913508, 7.77434848980775936382310046910, 8.380519276914765202130062799307

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