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Properties

Label 2-3724-931.778-c0-0-1
Degree $2$
Conductor $3724$
Sign $0.747 + 0.664i$
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  + (−1.72 − 0.829i)5-s + (0.826 − 0.563i)7-s + (0.623 + 0.781i)9-s + (0.455 − 0.571i)11-s + (−0.425 + 1.86i)17-s + 19-s + (−0.277 − 1.21i)23-s + (1.65 + 2.07i)25-s + (−1.88 + 0.284i)35-s + (1.32 − 0.636i)43-s + (−0.425 − 1.86i)45-s + (0.0931 − 0.116i)47-s + (0.365 − 0.930i)49-s + (−1.25 + 0.605i)55-s + (0.440 − 1.92i)61-s + ⋯
L(s)  = 1  + (−1.72 − 0.829i)5-s + (0.826 − 0.563i)7-s + (0.623 + 0.781i)9-s + (0.455 − 0.571i)11-s + (−0.425 + 1.86i)17-s + 19-s + (−0.277 − 1.21i)23-s + (1.65 + 2.07i)25-s + (−1.88 + 0.284i)35-s + (1.32 − 0.636i)43-s + (−0.425 − 1.86i)45-s + (0.0931 − 0.116i)47-s + (0.365 − 0.930i)49-s + (−1.25 + 0.605i)55-s + (0.440 − 1.92i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.747 + 0.664i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $0.747 + 0.664i$
Analytic conductor: \(1.85851\)
Root analytic conductor: \(1.36327\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (1709, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3724,\ (\ :0),\ 0.747 + 0.664i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.100527548\)
\(L(\frac12)\) \(\approx\) \(1.100527548\)
\(L(1)\) 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 \( 1 + (-0.826 + 0.563i)T \)
19 \( 1 - T \)
good3 \( 1 + (-0.623 - 0.781i)T^{2} \)
5 \( 1 + (1.72 + 0.829i)T + (0.623 + 0.781i)T^{2} \)
11 \( 1 + (-0.455 + 0.571i)T + (-0.222 - 0.974i)T^{2} \)
13 \( 1 + (0.222 + 0.974i)T^{2} \)
17 \( 1 + (0.425 - 1.86i)T + (-0.900 - 0.433i)T^{2} \)
23 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
29 \( 1 + (0.900 + 0.433i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.900 + 0.433i)T^{2} \)
41 \( 1 + (-0.623 - 0.781i)T^{2} \)
43 \( 1 + (-1.32 + 0.636i)T + (0.623 - 0.781i)T^{2} \)
47 \( 1 + (-0.0931 + 0.116i)T + (-0.222 - 0.974i)T^{2} \)
53 \( 1 + (0.900 - 0.433i)T^{2} \)
59 \( 1 + (-0.623 + 0.781i)T^{2} \)
61 \( 1 + (-0.440 + 1.92i)T + (-0.900 - 0.433i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.623 + 0.781i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \)
89 \( 1 + (0.222 - 0.974i)T^{2} \)
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

−8.444932149926985318008846781809, −7.896186067014359352898364782341, −7.46397420196781494027024754595, −6.54088680021669013902593557596, −5.39233113041657909413660259061, −4.55340287337007024138580923613, −4.14449429714216994389329980668, −3.47314099150125983143771791820, −1.85875320880209357719685777199, −0.851821009483914572553389300463, 1.06347241349444962190911334197, 2.56509965830045037364162214834, 3.36216880356414675983666453834, 4.20503591908489484306301704617, 4.75003142169275200115059397068, 5.81515828758324831611892471549, 6.99163283157278461754382192617, 7.28589425785876073641677637683, 7.77861538394966348227325256562, 8.784728385810212033620821254653

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