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Properties

Label 2-3800-152.11-c0-0-3
Degree $2$
Conductor $3800$
Sign $0.211 + 0.977i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s − 0.999i·8-s + (0.5 + 0.866i)9-s − 11-s + (−1.73 + i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 0.999i·18-s + (0.5 − 0.866i)19-s + (0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + 1.99·26-s + (0.866 − 0.499i)28-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s − i·7-s − 0.999i·8-s + (0.5 + 0.866i)9-s − 11-s + (−1.73 + i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 0.999i·18-s + (0.5 − 0.866i)19-s + (0.866 + 0.5i)22-s + (0.866 − 0.5i)23-s + 1.99·26-s + (0.866 − 0.499i)28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 + 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.211 + 0.977i$
Analytic conductor: \(1.89644\)
Root analytic conductor: \(1.37711\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3800} (3051, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3800,\ (\ :0),\ 0.211 + 0.977i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6954844204\)
\(L(\frac12)\) \(\approx\) \(0.6954844204\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 \)
19 \( 1 + (-0.5 + 0.866i)T \)
good3 \( 1 + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + iT - T^{2} \)
11 \( 1 + T + T^{2} \)
13 \( 1 + (1.73 - i)T + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + iT - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T^{2} \)
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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

−8.586986878273112316803614494232, −7.61962768254435345878303803543, −7.21424719695008209312031702731, −6.91514912886485700019260863870, −5.31461634865850129201703975243, −4.63615018930000453173954688071, −3.85289990108179059349710392732, −2.56448657674258295298196079886, −2.12870397932169803519979809873, −0.60951523058337795660914733215, 1.06910423054643075536548621834, 2.46762127695803102155229364685, 2.97289878692596580322312483449, 4.53370993129849666548492710319, 5.53729289182900667681438338530, 5.66670327990020302100403005652, 6.85249062394526421581441082515, 7.50576138107761045141709849774, 7.981277646816236938123839231039, 8.891247755919744119888976839759

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