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Properties

Label 2-3800-152.131-c0-0-0
Degree $2$
Conductor $3800$
Sign $0.944 - 0.327i$
Analytic cond. $1.89644$
Root an. cond. $1.37711$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (−1.17 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)6-s + (0.5 − 0.866i)8-s + (0.233 + 1.32i)9-s + (−0.173 + 0.300i)11-s + (0.766 + 1.32i)12-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s − 1.34·18-s + (0.173 + 0.984i)19-s + (−0.266 − 0.223i)22-s + (−1.43 + 0.524i)24-s + (0.266 − 0.460i)27-s + ⋯
L(s)  = 1  + (−0.173 + 0.984i)2-s + (−1.17 − 0.984i)3-s + (−0.939 − 0.342i)4-s + (1.17 − 0.984i)6-s + (0.5 − 0.866i)8-s + (0.233 + 1.32i)9-s + (−0.173 + 0.300i)11-s + (0.766 + 1.32i)12-s + (0.766 + 0.642i)16-s + (0.173 − 0.984i)17-s − 1.34·18-s + (0.173 + 0.984i)19-s + (−0.266 − 0.223i)22-s + (−1.43 + 0.524i)24-s + (0.266 − 0.460i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (1651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.944 - 0.327i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6205534919\)
\(L(\frac12)\) \(\approx\) \(0.6205534919\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 - 0.984i)T \)
5 \( 1 \)
19 \( 1 + (-0.173 - 0.984i)T \)
good3 \( 1 + (1.17 + 0.984i)T + (0.173 + 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.173 - 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.43 - 1.20i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (-1.53 + 1.28i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
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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

−8.378723218672269852918336249285, −7.67505223624728593880044175069, −7.26447309271366466477890311926, −6.38952064675435044293764398124, −6.03263129477704931778442171559, −5.15053589986988749687109491391, −4.65588623403090266911680122197, −3.37616031694706102480738190385, −1.83259384862959404810141391154, −0.75136841781040699102854412938, 0.73053505432368097083646628390, 2.15419658212726388349431128931, 3.32494972547290781847087915164, 4.04082048391047313774880645067, 4.77301017426330566594419210456, 5.42887709565153742473196709767, 6.10167510052054561002321645726, 7.16262714705482643793832056745, 8.178643360137083862704535102446, 8.904204442937355220919729505870

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