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Properties

Label 2-3724-931.816-c0-0-0
Degree $2$
Conductor $3724$
Sign $0.871 - 0.490i$
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  + (−0.914 + 0.848i)5-s + (0.955 + 0.294i)7-s + (0.826 + 0.563i)9-s + (1.36 − 0.930i)11-s + (0.266 + 0.680i)17-s + (−0.5 − 0.866i)19-s + (0.603 − 1.53i)23-s + (0.0414 − 0.553i)25-s + (−1.12 + 0.541i)35-s + (−0.277 + 1.21i)43-s + (−1.23 + 0.185i)45-s + (−0.109 − 1.46i)47-s + (0.826 + 0.563i)49-s + (−0.458 + 2.00i)55-s + (−0.147 − 0.0222i)61-s + ⋯
L(s)  = 1  + (−0.914 + 0.848i)5-s + (0.955 + 0.294i)7-s + (0.826 + 0.563i)9-s + (1.36 − 0.930i)11-s + (0.266 + 0.680i)17-s + (−0.5 − 0.866i)19-s + (0.603 − 1.53i)23-s + (0.0414 − 0.553i)25-s + (−1.12 + 0.541i)35-s + (−0.277 + 1.21i)43-s + (−1.23 + 0.185i)45-s + (−0.109 − 1.46i)47-s + (0.826 + 0.563i)49-s + (−0.458 + 2.00i)55-s + (−0.147 − 0.0222i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (3609, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.871 - 0.490i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.436801660\)
\(L(\frac12)\) \(\approx\) \(1.436801660\)
\(L(1)\) 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 \( 1 + (-0.955 - 0.294i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
good3 \( 1 + (-0.826 - 0.563i)T^{2} \)
5 \( 1 + (0.914 - 0.848i)T + (0.0747 - 0.997i)T^{2} \)
11 \( 1 + (-1.36 + 0.930i)T + (0.365 - 0.930i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.266 - 0.680i)T + (-0.733 + 0.680i)T^{2} \)
23 \( 1 + (-0.603 + 1.53i)T + (-0.733 - 0.680i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.955 - 0.294i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.109 + 1.46i)T + (-0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.955 + 0.294i)T^{2} \)
59 \( 1 + (-0.0747 - 0.997i)T^{2} \)
61 \( 1 + (0.147 + 0.0222i)T + (0.955 + 0.294i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (0.0747 - 0.997i)T + (-0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.658 - 0.317i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (-0.365 - 0.930i)T^{2} \)
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

−8.394055954942170782940489904103, −8.275052750440290925003953550378, −7.12669967909171973597175991034, −6.78436043734147236162400233402, −5.87812873421350026655562666830, −4.74252609466834948633934226515, −4.19627721565754872507008786378, −3.37626912677244997486844241385, −2.34251241415137264576270793660, −1.19281341270905322285417476442, 1.13607967126861699214337220069, 1.73174979884162949069378551601, 3.48972103652971538141167284287, 4.17497812640936550742101015456, 4.58846750587244601167155377933, 5.47565556046807491325534399592, 6.61389980757906712242798718591, 7.34773725621237009815356877787, 7.76264084013015765664401300209, 8.657896214486806733375150324163

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