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Properties

Label 2-3724-931.436-c0-0-0
Degree $2$
Conductor $3724$
Sign $-0.949 - 0.315i$
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  + (−0.109 + 1.46i)5-s + (−0.900 + 0.433i)7-s + (0.365 + 0.930i)9-s + (0.455 − 1.16i)11-s + (−1.40 + 1.29i)17-s + (−0.5 + 0.866i)19-s + (−0.535 − 0.496i)23-s + (−1.13 − 0.171i)25-s + (−0.535 − 1.36i)35-s + (−1.72 − 0.829i)43-s + (−1.40 + 0.432i)45-s + (1.78 − 0.268i)47-s + (0.623 − 0.781i)49-s + (1.64 + 0.793i)55-s + (1.19 + 0.367i)61-s + ⋯
L(s)  = 1  + (−0.109 + 1.46i)5-s + (−0.900 + 0.433i)7-s + (0.365 + 0.930i)9-s + (0.455 − 1.16i)11-s + (−1.40 + 1.29i)17-s + (−0.5 + 0.866i)19-s + (−0.535 − 0.496i)23-s + (−1.13 − 0.171i)25-s + (−0.535 − 1.36i)35-s + (−1.72 − 0.829i)43-s + (−1.40 + 0.432i)45-s + (1.78 − 0.268i)47-s + (0.623 − 0.781i)49-s + (1.64 + 0.793i)55-s + (1.19 + 0.367i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.315i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.949 - 0.315i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (3229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ -0.949 - 0.315i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7137107258\)
\(L(\frac12)\) \(\approx\) \(0.7137107258\)
\(L(1)\) 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 \( 1 + (0.900 - 0.433i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.365 - 0.930i)T^{2} \)
5 \( 1 + (0.109 - 1.46i)T + (-0.988 - 0.149i)T^{2} \)
11 \( 1 + (-0.455 + 1.16i)T + (-0.733 - 0.680i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (1.40 - 1.29i)T + (0.0747 - 0.997i)T^{2} \)
23 \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.826 - 0.563i)T^{2} \)
41 \( 1 + (-0.623 + 0.781i)T^{2} \)
43 \( 1 + (1.72 + 0.829i)T + (0.623 + 0.781i)T^{2} \)
47 \( 1 + (-1.78 + 0.268i)T + (0.955 - 0.294i)T^{2} \)
53 \( 1 + (-0.826 + 0.563i)T^{2} \)
59 \( 1 + (0.988 - 0.149i)T^{2} \)
61 \( 1 + (-1.19 - 0.367i)T + (0.826 + 0.563i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.900 + 0.433i)T^{2} \)
73 \( 1 + (1.97 + 0.298i)T + (0.955 + 0.294i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.914 - 1.14i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.733 - 0.680i)T^{2} \)
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

−8.733816088153255774520067036561, −8.479519301126154577843604438093, −7.43166160362749635226407274303, −6.65101996479075985949730215520, −6.24936475616732154453383069491, −5.56628044551996701027822282647, −4.14189059143065900667319945570, −3.63160119499145956504063561616, −2.65899609260268344223543869833, −1.94945831837895790343847654678, 0.39244817662955645289995198493, 1.56465631318067239855286659987, 2.78296147793536805615972380305, 4.06008593857096166952233209463, 4.39430118510698688722182990151, 5.17265396210580929283830367715, 6.30526079081185385394328337247, 6.91036443293047457247662042505, 7.45427908966747987567576936973, 8.692561415991109523236308608743

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