↑

Properties

Label 1-3724-3724.27-r1-0-0
Degree $1$
Conductor $3724$
Sign $-0.333 - 0.942i$
Analytic cond. $400.199$
Root an. cond. $400.199$
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  + (−0.0747 − 0.997i)3-s + (−0.0747 − 0.997i)5-s + (−0.988 + 0.149i)9-s + (−0.623 + 0.781i)11-s + (0.365 + 0.930i)13-s + (−0.988 + 0.149i)15-s + (−0.955 + 0.294i)17-s + (−0.955 − 0.294i)23-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (0.733 + 0.680i)29-s − 31-s + (0.826 + 0.563i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯
L(s)  = 1  + (−0.0747 − 0.997i)3-s + (−0.0747 − 0.997i)5-s + (−0.988 + 0.149i)9-s + (−0.623 + 0.781i)11-s + (0.365 + 0.930i)13-s + (−0.988 + 0.149i)15-s + (−0.955 + 0.294i)17-s + (−0.955 − 0.294i)23-s + (−0.988 + 0.149i)25-s + (0.222 + 0.974i)27-s + (0.733 + 0.680i)29-s − 31-s + (0.826 + 0.563i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.333 - 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3724\)    =    \(2^{2} \cdot 7^{2} \cdot 19\)
Sign: $-0.333 - 0.942i$
Analytic conductor: \(400.199\)
Root analytic conductor: \(400.199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3724} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3724,\ (1:\ ),\ -0.333 - 0.942i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3898384338 - 0.5515394317i\)
\(L(\frac12)\) \(\approx\) \(0.3898384338 - 0.5515394317i\)
\(L(1)\) \(\approx\) \(0.7177583411 - 0.2558545364i\)
\(L(1)\) \(\approx\) \(0.7177583411 - 0.2558545364i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
19 \( 1 \)
good3 \( 1 + (-0.0747 - 0.997i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (-0.623 + 0.781i)T \)
13 \( 1 + (0.365 + 0.930i)T \)
17 \( 1 + (-0.955 + 0.294i)T \)
23 \( 1 + (-0.955 - 0.294i)T \)
29 \( 1 + (0.733 + 0.680i)T \)
31 \( 1 - T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (0.0747 + 0.997i)T \)
43 \( 1 + (-0.826 - 0.563i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (-0.955 - 0.294i)T \)
59 \( 1 + (-0.826 - 0.563i)T \)
61 \( 1 + (0.733 + 0.680i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.733 + 0.680i)T \)
73 \( 1 + (-0.365 + 0.930i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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

−18.491002445931663098441645785059, −18.012320335278943863738160152213, −17.355048801311884763468612159571, −16.44295451718195169979569839044, −15.734368283245202812487281681905, −15.39348200666648438705977796135, −14.733394478509100708648236874184, −13.756091045685293319906083222410, −13.54445034429383038416485686488, −12.30413384403182605149827907948, −11.41178128121004019240875131676, −10.98827150336775993211988877076, −10.33875842720329681005220454879, −9.84747515987759102083981419732, −8.8560673892145850827467330931, −8.1875639181584333284839067716, −7.49421138486826528590427027698, −6.35859913071752252116876405227, −5.91238407490330657537053245615, −5.09001979606812040100107671756, −4.20986386187555168047695516796, −3.36040358851697862853317881434, −2.93801961911755077250849069291, −2.0202407588552399523092682406, −0.41755112767058593845485787481, 0.202104883397004632755100837147, 1.39514886287391442886127632884, 1.87277065532743275498407866517, 2.70314439077613575600068820002, 3.98319817215755839290030553054, 4.621420335654398615782213746451, 5.45025117133787731749104422686, 6.219796447332322123490471872, 6.95543644475645863254505919341, 7.673799892262067656587761260495, 8.42587853150968071593049833246, 8.93417641078903904148735925995, 9.74035349042168419167448181135, 10.75743502038289766181726460921, 11.49655198237407374968038126413, 12.16242076960123310097914800460, 12.83330047768846066404608604579, 13.162867114154187107647965004992, 14.037922707689954913328732586261, 14.63444120435194117320897739207, 15.728385259557795100293122829421, 16.2174161847330156784968668965, 16.94063033236553191914432152203, 17.76446244299978788900123326584, 18.064245516327468184490101958439

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