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Properties

Label 2-3724-931.379-c0-0-1
Degree $2$
Conductor $3724$
Sign $0.656 + 0.754i$
Analytic cond. $1.85851$
Root an. cond. $1.36327$
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  + (0.0931 + 0.116i)5-s + (−0.988 + 0.149i)7-s + (−0.222 + 0.974i)9-s + (−0.425 − 1.86i)11-s + (−0.134 − 0.0648i)17-s + 19-s + (0.400 − 0.193i)23-s + (0.217 − 0.953i)25-s + (−0.109 − 0.101i)35-s + (1.03 − 1.29i)43-s + (−0.134 + 0.0648i)45-s + (−0.162 − 0.712i)47-s + (0.955 − 0.294i)49-s + (0.178 − 0.223i)55-s + (1.32 + 0.636i)61-s + ⋯
L(s)  = 1  + (0.0931 + 0.116i)5-s + (−0.988 + 0.149i)7-s + (−0.222 + 0.974i)9-s + (−0.425 − 1.86i)11-s + (−0.134 − 0.0648i)17-s + 19-s + (0.400 − 0.193i)23-s + (0.217 − 0.953i)25-s + (−0.109 − 0.101i)35-s + (1.03 − 1.29i)43-s + (−0.134 + 0.0648i)45-s + (−0.162 − 0.712i)47-s + (0.955 − 0.294i)49-s + (0.178 − 0.223i)55-s + (1.32 + 0.636i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.656 + 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.656 + 0.754i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (2241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.656 + 0.754i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.016537633\)
\(L(\frac12)\) \(\approx\) \(1.016537633\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (0.988 - 0.149i)T \)
19 \( 1 - T \)
good3 \( 1 + (0.222 - 0.974i)T^{2} \)
5 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (0.425 + 1.86i)T + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.134 + 0.0648i)T + (0.623 + 0.781i)T^{2} \)
23 \( 1 + (-0.400 + 0.193i)T + (0.623 - 0.781i)T^{2} \)
29 \( 1 + (-0.623 - 0.781i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-0.623 - 0.781i)T^{2} \)
41 \( 1 + (0.222 - 0.974i)T^{2} \)
43 \( 1 + (-1.03 + 1.29i)T + (-0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 + (0.222 + 0.974i)T^{2} \)
61 \( 1 + (-1.32 - 0.636i)T + (0.623 + 0.781i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.222 + 0.974i)T + (-0.900 - 0.433i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.900 + 0.433i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.665679016840722218740033183644, −7.913734119809002161657997096959, −7.13851772352496019436766171552, −6.24775350231454796652981661732, −5.66453208770114357496938593510, −5.01303969568948028731857199465, −3.76700027175034710459318774901, −3.02832674226625929604360748456, −2.35543601530567826353435013950, −0.65307929586574015154308353623, 1.18161836097918337504219112037, 2.48973480870438602465722358417, 3.31363354035087557205956203754, 4.15729198143902906188487658919, 5.05058884150869365134195083757, 5.83425331280718561637177796052, 6.76506751901477156795316925191, 7.15724707392678473172407714555, 7.940220710824742279353123307076, 9.071049727650734707451889580149

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