Properties

Label 1-4011-4011.353-r0-0-0
Degree $1$
Conductor $4011$
Sign $0.419 + 0.907i$
Analytic cond. $18.6270$
Root an. cond. $18.6270$
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  + (−0.988 + 0.153i)2-s + (0.952 − 0.303i)4-s + (0.993 + 0.110i)5-s + (−0.894 + 0.446i)8-s + (−0.999 + 0.0440i)10-s + (−0.904 + 0.426i)11-s + (0.956 − 0.293i)13-s + (0.815 − 0.578i)16-s + (−0.381 − 0.924i)17-s + (−0.999 − 0.0440i)19-s + (0.980 − 0.197i)20-s + (0.828 − 0.560i)22-s + (0.782 + 0.622i)23-s + (0.975 + 0.218i)25-s + (−0.899 + 0.436i)26-s + ⋯
L(s)  = 1  + (−0.988 + 0.153i)2-s + (0.952 − 0.303i)4-s + (0.993 + 0.110i)5-s + (−0.894 + 0.446i)8-s + (−0.999 + 0.0440i)10-s + (−0.904 + 0.426i)11-s + (0.956 − 0.293i)13-s + (0.815 − 0.578i)16-s + (−0.381 − 0.924i)17-s + (−0.999 − 0.0440i)19-s + (0.980 − 0.197i)20-s + (0.828 − 0.560i)22-s + (0.782 + 0.622i)23-s + (0.975 + 0.218i)25-s + (−0.899 + 0.436i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4011 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.419 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4011\)    =    \(3 \cdot 7 \cdot 191\)
Sign: $0.419 + 0.907i$
Analytic conductor: \(18.6270\)
Root analytic conductor: \(18.6270\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4011} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4011,\ (0:\ ),\ 0.419 + 0.907i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9672522509 + 0.6185841419i\)
\(L(\frac12)\) \(\approx\) \(0.9672522509 + 0.6185841419i\)
\(L(1)\) \(\approx\) \(0.8014817504 + 0.1245745560i\)
\(L(1)\) \(\approx\) \(0.8014817504 + 0.1245745560i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
191 \( 1 \)
good2 \( 1 + (-0.988 + 0.153i)T \)
5 \( 1 + (0.993 + 0.110i)T \)
11 \( 1 + (-0.904 + 0.426i)T \)
13 \( 1 + (0.956 - 0.293i)T \)
17 \( 1 + (-0.381 - 0.924i)T \)
19 \( 1 + (-0.999 - 0.0440i)T \)
23 \( 1 + (0.782 + 0.622i)T \)
29 \( 1 + (-0.115 - 0.993i)T \)
31 \( 1 + (0.926 + 0.376i)T \)
37 \( 1 + (-0.926 + 0.376i)T \)
41 \( 1 + (0.546 + 0.837i)T \)
43 \( 1 + (-0.768 + 0.639i)T \)
47 \( 1 + (0.329 + 0.944i)T \)
53 \( 1 + (0.126 + 0.991i)T \)
59 \( 1 + (-0.644 + 0.764i)T \)
61 \( 1 + (-0.761 - 0.648i)T \)
67 \( 1 + (0.565 - 0.824i)T \)
71 \( 1 + (-0.0495 + 0.998i)T \)
73 \( 1 + (-0.999 - 0.0220i)T \)
79 \( 1 + (0.802 + 0.596i)T \)
83 \( 1 + (-0.148 - 0.988i)T \)
89 \( 1 + (0.618 + 0.785i)T \)
97 \( 1 + (0.999 + 0.0330i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.35980973462980874040608211957, −17.718619763068071475952969978121, −17.04468919556723025255764167056, −16.58037680388821989760117975068, −15.77640518048720220113341681032, −15.15162783929271758810078979530, −14.31499969251549689676822958438, −13.35090407190772071727463177419, −12.94422187796126505618450490323, −12.16956968836567873307297881119, −11.11941581074458147086480924884, −10.48736874503991058240869614279, −10.358348110518973305019908457689, −9.15230723967986875048201944667, −8.66890923292211632252644278436, −8.249727373184933665232264268192, −7.093922812236904537442621752591, −6.45201293148419033222982870737, −5.88181432167294815924654478643, −5.032289006353280725920831629070, −3.86120992772768644734066646555, −2.99232949421930746854564587260, −2.13378128181944174927849402345, −1.60194281887607632511164198731, −0.498161815809909517515659127360, 0.92605199491862952976069153998, 1.72872695665372793399013174906, 2.59888733891550335532969579272, 3.0847540025754226388449507747, 4.5678076929678888683207174591, 5.328148253972135047262572617414, 6.15762901671740862756896098147, 6.61514083113860000738028528011, 7.52209896343799785541422851152, 8.19370043557643076506702518953, 9.001976964873805491051076988682, 9.52477926229609142907711162258, 10.31169046800352442964327521027, 10.769785136850604541438252459541, 11.45613250076600763621259287102, 12.419791468754070311691479423237, 13.26493921425016583061196402182, 13.712483601580196212816155702788, 14.71618107712472795176998290993, 15.44675324489583674732092527929, 15.84959405773172716425817514937, 16.761542326109808445662992229079, 17.45882205977279259463949778630, 17.784811789887735309753112160189, 18.61264881579397289692423815292

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