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Properties

Label 2-3724-931.170-c0-0-0
Degree $2$
Conductor $3724$
Sign $-0.0106 - 0.999i$
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  + (−0.535 − 0.496i)5-s + (−0.222 + 0.974i)7-s + (0.826 − 0.563i)9-s + (−1.48 − 1.01i)11-s + (−0.722 + 1.84i)17-s + (−0.5 + 0.866i)19-s + (0.603 + 1.53i)23-s + (−0.0348 − 0.464i)25-s + (0.603 − 0.411i)35-s + (0.440 + 1.92i)43-s + (−0.722 − 0.108i)45-s + (−0.0332 + 0.443i)47-s + (−0.900 − 0.433i)49-s + (0.292 + 1.28i)55-s + (1.78 − 0.268i)61-s + ⋯
L(s)  = 1  + (−0.535 − 0.496i)5-s + (−0.222 + 0.974i)7-s + (0.826 − 0.563i)9-s + (−1.48 − 1.01i)11-s + (−0.722 + 1.84i)17-s + (−0.5 + 0.866i)19-s + (0.603 + 1.53i)23-s + (−0.0348 − 0.464i)25-s + (0.603 − 0.411i)35-s + (0.440 + 1.92i)43-s + (−0.722 − 0.108i)45-s + (−0.0332 + 0.443i)47-s + (−0.900 − 0.433i)49-s + (0.292 + 1.28i)55-s + (1.78 − 0.268i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0106 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.0106 - 0.999i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ -0.0106 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7313554010\)
\(L(\frac12)\) \(\approx\) \(0.7313554010\)
\(L(1)\) 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 \( 1 + (-0.826 + 0.563i)T^{2} \)
5 \( 1 + (0.535 + 0.496i)T + (0.0747 + 0.997i)T^{2} \)
11 \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (0.722 - 1.84i)T + (-0.733 - 0.680i)T^{2} \)
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 \( 1 + (-0.440 - 1.92i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (0.0332 - 0.443i)T + (-0.988 - 0.149i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (-1.78 + 0.268i)T + (0.955 - 0.294i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.149 - 1.99i)T + (-0.988 + 0.149i)T^{2} \)
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

−8.631722231640832986923549981601, −8.270933270644726077734428101300, −7.64969396256853439442663517029, −6.49142654610266296251837472035, −5.90825678665498423666735361673, −5.22140672967732447950285294100, −4.20132805317639194232536396828, −3.52919663222828319762714682093, −2.52046774174746375665532139051, −1.36787123244115362991737371481, 0.42292451016538524003399470322, 2.21346090922106945398782406575, 2.81966735511689940315961441712, 4.02720222044427490198263487310, 4.74053356987095331017955821969, 5.16099181157061137482797201679, 6.78026663658187042748954912256, 7.11053389041571760881069631371, 7.46564463400252218385987456747, 8.374825559530977775244623930424

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