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Properties

Label 2-3800-152.11-c0-0-4
Degree $2$
Conductor $3800$
Sign $0.977 - 0.211i$
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  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + 2·11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−1 + 1.73i)22-s + (0.5 − 0.866i)24-s + 27-s + (−0.499 − 0.866i)32-s + (1 − 1.73i)33-s + (0.499 + 0.866i)34-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 − 0.866i)4-s + (0.499 + 0.866i)6-s + 0.999·8-s + 2·11-s − 0.999·12-s + (−0.5 + 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (−1 + 1.73i)22-s + (0.5 − 0.866i)24-s + 27-s + (−0.499 − 0.866i)32-s + (1 − 1.73i)33-s + (0.499 + 0.866i)34-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $0.977 - 0.211i$
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: \((2,\ 3800,\ (\ :0),\ 0.977 - 0.211i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.335557611\)
\(L(\frac12)\) \(\approx\) \(1.335557611\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

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

−8.441821272299487408350601579771, −8.001334529556318652328485316867, −7.24693337948774978607277028341, −6.54846842392053870969302549850, −6.21749388107057116802928548123, −5.03274828592602820856014622212, −4.26049770546453048896385894453, −3.20578092852469032848447804537, −1.78856357861335824141158872768, −1.20097690071042447793271626732, 1.14076181499277350436845037254, 2.17710874215131130382050232223, 3.34872671196774345891539005319, 3.91308153171926891045136119018, 4.33534633561389999883073065349, 5.48559222215828681291632192774, 6.69682002586442432965200899465, 7.18566393109695194420000894650, 8.513231889612809252378669052255, 8.792659989163218011338691669875

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