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Properties

Label 2-3751-341.28-c0-0-0
Degree $2$
Conductor $3751$
Sign $0.251 - 0.967i$
Analytic cond. $1.87199$
Root an. cond. $1.36820$
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  + (0.413 − 0.459i)3-s + (0.309 + 0.951i)4-s + (1.22 + 1.35i)5-s + (0.0646 + 0.614i)9-s + (0.564 + 0.251i)12-s + 1.12·15-s + (−0.809 + 0.587i)16-s + (−0.913 + 1.58i)20-s + (−0.604 − 1.86i)23-s + (−0.244 + 2.32i)25-s + (0.809 + 0.587i)27-s + (0.669 − 0.743i)31-s + (−0.564 + 0.251i)36-s + (−0.913 − 0.406i)37-s + (−0.755 + 0.839i)45-s + ⋯
L(s)  = 1  + (0.413 − 0.459i)3-s + (0.309 + 0.951i)4-s + (1.22 + 1.35i)5-s + (0.0646 + 0.614i)9-s + (0.564 + 0.251i)12-s + 1.12·15-s + (−0.809 + 0.587i)16-s + (−0.913 + 1.58i)20-s + (−0.604 − 1.86i)23-s + (−0.244 + 2.32i)25-s + (0.809 + 0.587i)27-s + (0.669 − 0.743i)31-s + (−0.564 + 0.251i)36-s + (−0.913 − 0.406i)37-s + (−0.755 + 0.839i)45-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.251 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.251 - 0.967i$
Analytic conductor: \(1.87199\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (2756, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3751,\ (\ :0),\ 0.251 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.971762574\)
\(L(\frac12)\) \(\approx\) \(1.971762574\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 + (-0.669 + 0.743i)T \)
good2 \( 1 + (-0.309 - 0.951i)T^{2} \)
3 \( 1 + (-0.413 + 0.459i)T + (-0.104 - 0.994i)T^{2} \)
5 \( 1 + (-1.22 - 1.35i)T + (-0.104 + 0.994i)T^{2} \)
7 \( 1 + (0.104 + 0.994i)T^{2} \)
13 \( 1 + (0.104 - 0.994i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.604 + 1.86i)T + (-0.809 + 0.587i)T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.913 + 0.406i)T + (0.669 + 0.743i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.669 - 0.743i)T^{2} \)
47 \( 1 + (0.0646 + 0.198i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.190 + 1.81i)T + (-0.978 + 0.207i)T^{2} \)
59 \( 1 + (-0.978 - 1.69i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.104 + 0.181i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.913 - 0.406i)T^{2} \)
79 \( 1 + (-0.669 - 0.743i)T^{2} \)
83 \( 1 + (-0.913 - 0.406i)T^{2} \)
89 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
97 \( 1 + 1.95T + T^{2} \)
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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

−8.573324697299657587151987078437, −8.134661474665390528593939036970, −7.19976956919875839254788592418, −6.79130036606368650649797551454, −6.18075177841986143475855528839, −5.18604520664589927862488191502, −4.03693083614270919044975188364, −3.06309658163754184813073876276, −2.37894099430592861079174342301, −1.97213290299597616992796405645, 1.12422694527807763798033292267, 1.76063383302263814277531764516, 2.88939254680745329382730644519, 4.07615510421349859112104190066, 4.91599561715441054851276421255, 5.51440886307189796147504497555, 6.11134847547232520386619280502, 6.85353617185717670626558258017, 8.014181344734240628332128017951, 8.872086624136227861683378001056

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