↑

Properties

Label 2-3751-341.247-c0-0-7
Degree $2$
Conductor $3751$
Sign $0.836 + 0.548i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)5-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)35-s + (0.309 − 0.951i)38-s + (0.809 − 0.587i)40-s + (0.809 + 0.587i)41-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (−0.309 + 0.951i)5-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + 0.999·10-s + (−0.809 − 0.587i)14-s + (−0.309 + 0.951i)16-s + (0.809 − 0.587i)18-s + (0.809 + 0.587i)19-s + (0.309 + 0.951i)31-s + (0.309 + 0.951i)35-s + (0.309 − 0.951i)38-s + (0.809 − 0.587i)40-s + (0.809 + 0.587i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.836 + 0.548i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.836 + 0.548i$
Analytic conductor: \(1.87199\)
Root analytic conductor: \(1.36820\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (3657, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3751,\ (\ :0),\ 0.836 + 0.548i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.235080702\)
\(L(\frac12)\) \(\approx\) \(1.235080702\)
\(L(1)\) 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 \( 1 + (-0.309 - 0.951i)T \)
good2 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.809 - 0.587i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.809 + 0.587i)T + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.570888773649916038116924941279, −7.918458218246660975820131485014, −7.17788716692682389545206295725, −6.64866357444128535494811275334, −5.54269249961013158993038487508, −4.70331804571578648177658197577, −3.71081662311317404951861877248, −3.00474460210756939814824299531, −2.06367105934054546589829418266, −1.22847165823830989185475726513, 0.914250522852303526762741448742, 2.25432362148014818247512297700, 3.34719726300541594378137495783, 4.43977330314896515654059069140, 5.13658670629985180458445421846, 5.84903157242174094247368260079, 6.61609867725206849535638733530, 7.37869396276421661573351592052, 8.092085653590668629332975512878, 8.544008727522573718119957580667

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