↑

Properties

Label 1-3751-3751.681-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.898 - 0.438i$
Analytic cond. $17.4195$
Root an. cond. $17.4195$
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  + (0.142 − 0.989i)2-s − 3-s + (−0.959 − 0.281i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (−0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + 9-s + (0.959 − 0.281i)10-s + (0.959 + 0.281i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)14-s + (−0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (−0.654 + 0.755i)17-s + (0.142 − 0.989i)18-s + ⋯
L(s)  = 1  + (0.142 − 0.989i)2-s − 3-s + (−0.959 − 0.281i)4-s + (0.415 + 0.909i)5-s + (−0.142 + 0.989i)6-s + (−0.841 + 0.540i)7-s + (−0.415 + 0.909i)8-s + 9-s + (0.959 − 0.281i)10-s + (0.959 + 0.281i)12-s + (−0.959 − 0.281i)13-s + (0.415 + 0.909i)14-s + (−0.415 − 0.909i)15-s + (0.841 + 0.540i)16-s + (−0.654 + 0.755i)17-s + (0.142 − 0.989i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.898 - 0.438i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.898 - 0.438i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (681, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.898 - 0.438i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6538977958 - 0.1511629082i\)
\(L(\frac12)\) \(\approx\) \(0.6538977958 - 0.1511629082i\)
\(L(1)\) \(\approx\) \(0.6168782890 - 0.1680409474i\)
\(L(1)\) \(\approx\) \(0.6168782890 - 0.1680409474i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (0.142 - 0.989i)T \)
3 \( 1 - T \)
5 \( 1 + (0.415 + 0.909i)T \)
7 \( 1 + (-0.841 + 0.540i)T \)
13 \( 1 + (-0.959 - 0.281i)T \)
17 \( 1 + (-0.654 + 0.755i)T \)
19 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (-0.841 - 0.540i)T \)
29 \( 1 + (-0.654 - 0.755i)T \)
37 \( 1 + (0.959 - 0.281i)T \)
41 \( 1 + (0.142 - 0.989i)T \)
43 \( 1 + (0.415 - 0.909i)T \)
47 \( 1 + (-0.142 - 0.989i)T \)
53 \( 1 + (-0.841 + 0.540i)T \)
59 \( 1 + (-0.142 - 0.989i)T \)
61 \( 1 + (-0.142 - 0.989i)T \)
67 \( 1 + (-0.142 + 0.989i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (0.841 + 0.540i)T \)
79 \( 1 + (0.415 + 0.909i)T \)
83 \( 1 + (0.841 - 0.540i)T \)
89 \( 1 + (0.654 - 0.755i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.132255264041422114217471166127, −17.85933547580658870236867222979, −17.135776162639766543272767477333, −16.4460997968917722468664395664, −16.237875318255202569529848751872, −15.5661296856991891279565432374, −14.608744802176604517894856344526, −13.62493556426015987909865818047, −13.28160279781327262419285203433, −12.5993571241683725951126290549, −11.98007171266531100007428879371, −11.110778661107912043774864760783, −9.96111023501014718835544430590, −9.52600467867330078106986987454, −9.10367266679580537495850265008, −7.73592723522566898470212599469, −7.34486966377926549598719336262, −6.43650442317911697826517745341, −6.01650209820658441444183632531, −5.02013940672842991382991719103, −4.71584473029788667138035075913, −3.93632745209012763398276837859, −2.76787728875026885796422317291, −1.35153164128619985365452677455, −0.45783437068379433009945364257, 0.47674710472702645758737217564, 1.96509890991944962939716255001, 2.284978251240252157469307820990, 3.40193020992730913533058241038, 4.00298650969202943597073815148, 5.01835328364840646425769074804, 5.866460185090630823551109659709, 6.12984972067702144343614570126, 7.13315661929469668424172929835, 8.00843191439657392859036048972, 9.229504121472526810655406640750, 9.81328350908202518933599385767, 10.27738127344735573998498605126, 10.90274454153864824051341369581, 11.67807829204598792641978894986, 12.29915128494192793175595977130, 12.79696384477045323863792259811, 13.543891651034151617514561912667, 14.351898064895204951657328381191, 15.11128099494991629452636351280, 15.7372988073870727129717765831, 16.77645252297472715453875004885, 17.39415241263006621229842888919, 17.98916314898019475544897548480, 18.72019275782003014158875867465

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