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Properties

Label 2-3800-152.11-c0-0-5
Degree $2$
Conductor $3800$
Sign $-0.305 + 0.952i$
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 − 11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)24-s − 27-s + (0.499 + 0.866i)32-s + (0.5 − 0.866i)33-s + (−0.999 − 1.73i)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 − 11-s + 0.999·12-s + (−0.5 + 0.866i)16-s + (1 − 1.73i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)22-s + (0.5 − 0.866i)24-s − 27-s + (0.499 + 0.866i)32-s + (0.5 − 0.866i)33-s + (−0.999 − 1.73i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3800\)    =    \(2^{3} \cdot 5^{2} \cdot 19\)
Sign: $-0.305 + 0.952i$
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: \((2,\ 3800,\ (\ :0),\ -0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.005381333\)
\(L(\frac12)\) \(\approx\) \(1.005381333\)
\(L(1)\) 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 \( 1 + T + T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
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

−8.801061667146172223900006302073, −7.67910070770663830479328246272, −6.94944209964752139845463856012, −5.66467494285632800173711656226, −5.32763178529595750794184896519, −4.67232190865074126792521798506, −3.91585151042771748337942254484, −2.92764953168571422288492782366, −2.20386241958171626950647796076, −0.54689964200041478865156603202, 1.30942933133815676285840403259, 2.61457635491670598605040696977, 3.69640326728159162261851970047, 4.40672017324912073846013203006, 5.58269055062843969766651745156, 5.91220156149461080410026856878, 6.53633365398466010734190753223, 7.46696163752520357293982429056, 7.908190608243251120909372638082, 8.455486710762704136513495198753

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