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Properties

Label 2-3800-760.579-c0-0-4
Degree $2$
Conductor $3800$
Sign $-0.269 + 0.962i$
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  + (−0.984 + 0.173i)2-s + (−0.223 − 0.266i)3-s + (0.939 − 0.342i)4-s + (0.266 + 0.223i)6-s + (−0.866 + 0.5i)8-s + (0.152 − 0.866i)9-s + (−0.766 − 1.32i)11-s + (−0.300 − 0.173i)12-s + (0.766 − 0.642i)16-s + (0.984 − 0.173i)17-s + 0.879i·18-s + (0.939 − 0.342i)19-s + (0.984 + 1.17i)22-s + (0.326 + 0.118i)24-s + (−0.565 + 0.326i)27-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (−0.223 − 0.266i)3-s + (0.939 − 0.342i)4-s + (0.266 + 0.223i)6-s + (−0.866 + 0.5i)8-s + (0.152 − 0.866i)9-s + (−0.766 − 1.32i)11-s + (−0.300 − 0.173i)12-s + (0.766 − 0.642i)16-s + (0.984 − 0.173i)17-s + 0.879i·18-s + (0.939 − 0.342i)19-s + (0.984 + 1.17i)22-s + (0.326 + 0.118i)24-s + (−0.565 + 0.326i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.269 + 0.962i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (2099, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ -0.269 + 0.962i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6331352364\)
\(L(\frac12)\) \(\approx\) \(0.6331352364\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.984 - 0.173i)T \)
5 \( 1 \)
19 \( 1 + (-0.939 + 0.342i)T \)
good3 \( 1 + (0.223 + 0.266i)T + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.766 + 1.32i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.984 + 0.173i)T + (0.939 - 0.342i)T^{2} \)
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 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.342 + 0.939i)T + (-0.766 - 0.642i)T^{2} \)
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 \( 1 + (1.85 + 0.326i)T + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (0.984 + 1.17i)T + (-0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (1.62 + 0.939i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-1.85 + 0.326i)T + (0.939 - 0.342i)T^{2} \)
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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

−8.495555670225961143256663248362, −7.69402678960920817195550640520, −7.21484100616960492785117995063, −6.22554878743062777191438400650, −5.81039928834076124221013621580, −4.96732450257438675208858219585, −3.37992498846284845333709177218, −2.97933394302109351026709091660, −1.52223199081406560504811939273, −0.53962991395079388359665739647, 1.42726740209580225802880947143, 2.30348989084768658119683245839, 3.23449541182322448838020535910, 4.33558087019610867584980696507, 5.26923830907038760169398517124, 5.87023316575258315450579021009, 7.16259164559469897297994900375, 7.42291369668156580090265828368, 8.109613714161352420942798072427, 8.916325190481835045329240565727

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