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Properties

Label 2-3724-931.37-c0-0-1
Degree $2$
Conductor $3724$
Sign $-0.860 - 0.509i$
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  + (−1.88 − 0.582i)5-s + (−0.222 − 0.974i)7-s + (0.0747 − 0.997i)9-s + (−0.134 − 1.79i)11-s + (−0.722 − 0.108i)17-s + (−0.5 + 0.866i)19-s + (−0.147 + 0.0222i)23-s + (2.40 + 1.63i)25-s + (−0.147 + 1.97i)35-s + (−0.162 + 0.712i)43-s + (−0.722 + 1.84i)45-s + (−0.367 + 0.250i)47-s + (−0.900 + 0.433i)49-s + (−0.792 + 3.47i)55-s + (−0.658 − 1.67i)61-s + ⋯
L(s)  = 1  + (−1.88 − 0.582i)5-s + (−0.222 − 0.974i)7-s + (0.0747 − 0.997i)9-s + (−0.134 − 1.79i)11-s + (−0.722 − 0.108i)17-s + (−0.5 + 0.866i)19-s + (−0.147 + 0.0222i)23-s + (2.40 + 1.63i)25-s + (−0.147 + 1.97i)35-s + (−0.162 + 0.712i)43-s + (−0.722 + 1.84i)45-s + (−0.367 + 0.250i)47-s + (−0.900 + 0.433i)49-s + (−0.792 + 3.47i)55-s + (−0.658 − 1.67i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.860 - 0.509i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ -0.860 - 0.509i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3409695687\)
\(L(\frac12)\) \(\approx\) \(0.3409695687\)
\(L(1)\) 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 \( 1 + (0.222 + 0.974i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
good3 \( 1 + (-0.0747 + 0.997i)T^{2} \)
5 \( 1 + (1.88 + 0.582i)T + (0.826 + 0.563i)T^{2} \)
11 \( 1 + (0.134 + 1.79i)T + (-0.988 + 0.149i)T^{2} \)
13 \( 1 + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (0.722 + 0.108i)T + (0.955 + 0.294i)T^{2} \)
23 \( 1 + (0.147 - 0.0222i)T + (0.955 - 0.294i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.733 - 0.680i)T^{2} \)
41 \( 1 + (0.900 - 0.433i)T^{2} \)
43 \( 1 + (0.162 - 0.712i)T + (-0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.367 - 0.250i)T + (0.365 - 0.930i)T^{2} \)
53 \( 1 + (0.733 + 0.680i)T^{2} \)
59 \( 1 + (-0.826 + 0.563i)T^{2} \)
61 \( 1 + (0.658 + 1.67i)T + (-0.733 + 0.680i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.65 - 1.12i)T + (0.365 + 0.930i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-1.78 + 0.858i)T + (0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.988 + 0.149i)T^{2} \)
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

−8.129532935090953343563464630827, −7.80013798071438066499032314981, −6.76620358011139686492902750229, −6.26045477417767508505364677562, −5.07060366077818690997683893446, −4.15279606525514081257078210911, −3.68277557347672912404330776464, −3.13968528565881676592977050230, −1.09835653041142875388264226897, −0.22656467052416317549442946369, 2.10601854173799218251851323213, 2.71080239285978576055351894160, 3.83807627226859258924781078368, 4.60519660988279658733067414237, 5.04569911402577288854557605815, 6.43711090883986782426130957795, 7.08672699319013183403392578782, 7.58868383761075395190997963775, 8.307031428159935480290216721726, 8.916347003742340000604475041650

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