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Properties

Label 1-3751-3751.1101-r0-0-0
Degree $1$
Conductor $3751$
Sign $0.659 + 0.751i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.736 − 0.676i)2-s + (0.309 + 0.951i)3-s + (0.0855 + 0.996i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (−0.466 + 0.884i)7-s + (0.610 − 0.791i)8-s + (−0.809 + 0.587i)9-s + (0.516 + 0.856i)10-s + (−0.921 + 0.389i)12-s + (0.516 − 0.856i)13-s + (0.941 − 0.336i)14-s + (−0.0285 − 0.999i)15-s + (−0.985 + 0.170i)16-s + (−0.998 + 0.0570i)17-s + (0.993 + 0.113i)18-s + ⋯
L(s)  = 1  + (−0.736 − 0.676i)2-s + (0.309 + 0.951i)3-s + (0.0855 + 0.996i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (−0.466 + 0.884i)7-s + (0.610 − 0.791i)8-s + (−0.809 + 0.587i)9-s + (0.516 + 0.856i)10-s + (−0.921 + 0.389i)12-s + (0.516 − 0.856i)13-s + (0.941 − 0.336i)14-s + (−0.0285 − 0.999i)15-s + (−0.985 + 0.170i)16-s + (−0.998 + 0.0570i)17-s + (0.993 + 0.113i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3751 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.659 + 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3751\)    =    \(11^{2} \cdot 31\)
Sign: $0.659 + 0.751i$
Analytic conductor: \(17.4195\)
Root analytic conductor: \(17.4195\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3751} (1101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3751,\ (0:\ ),\ 0.659 + 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4787211966 + 0.2169051790i\)
\(L(\frac12)\) \(\approx\) \(0.4787211966 + 0.2169051790i\)
\(L(1)\) \(\approx\) \(0.5551239535 + 0.03565680240i\)
\(L(1)\) \(\approx\) \(0.5551239535 + 0.03565680240i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.736 - 0.676i)T \)
3 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 + (-0.959 - 0.281i)T \)
7 \( 1 + (-0.466 + 0.884i)T \)
13 \( 1 + (0.516 - 0.856i)T \)
17 \( 1 + (-0.998 + 0.0570i)T \)
19 \( 1 + (-0.254 - 0.967i)T \)
23 \( 1 + (-0.466 - 0.884i)T \)
29 \( 1 + (-0.254 - 0.967i)T \)
37 \( 1 + (-0.654 + 0.755i)T \)
41 \( 1 + (-0.736 - 0.676i)T \)
43 \( 1 + (-0.0285 + 0.999i)T \)
47 \( 1 + (-0.736 + 0.676i)T \)
53 \( 1 + (0.696 + 0.717i)T \)
59 \( 1 + (-0.736 + 0.676i)T \)
61 \( 1 + (0.415 + 0.909i)T \)
67 \( 1 + (0.415 - 0.909i)T \)
71 \( 1 + (-0.362 + 0.931i)T \)
73 \( 1 + (0.696 - 0.717i)T \)
79 \( 1 + (0.941 - 0.336i)T \)
83 \( 1 + (-0.985 - 0.170i)T \)
89 \( 1 + (-0.362 - 0.931i)T \)
97 \( 1 + (0.941 + 0.336i)T \)
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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

−18.54775662866764895780259730385, −17.96838288171071914290984290274, −17.14377826765115496014764857741, −16.50013653148709612423460903721, −15.91015399617402792262025012054, −15.16671474027391072442169406346, −14.37530045549333092596078671936, −13.88237035163963136590409534788, −13.19607052644666339023964312053, −12.28546667393451078156155489886, −11.42848534002188735688950382500, −10.95699234424276448271977777859, −10.07707195742652081344567505359, −9.1955614515548502331275740276, −8.48162872473925927169028077428, −7.95791718937276137005670463272, −7.09548137056237888128643571117, −6.84748691545976177798137604833, −6.180267903724223028996147781662, −5.10478234060710772209147596496, −3.93836196356645400428895523752, −3.46622596591756275197247245884, −2.10983555162417923362936344182, −1.44796045440015155363207452245, −0.35393473622823460288109252913, 0.52473904944445807608212497156, 2.01671765214747947046239021709, 2.84970014547015844055715133551, 3.3139579282221649076420845375, 4.264891961007955565419820575640, 4.73420086242455615502348025993, 5.88869306502570555014389813751, 6.82890996121233531340011442912, 7.87090248383747520368998676025, 8.55505452050870259185371296652, 8.78770177837642384063568826579, 9.60945061963314402938163988388, 10.36647462604717459963837598605, 11.055041847476653223166190420397, 11.56521493441748946135772594510, 12.33027981987254194373285064438, 13.03614881445419988074655153324, 13.70311977227907467147560662506, 15.13415979147109143910082926500, 15.3432932211075523529431229061, 15.963745925146156963027639555564, 16.556174395721253409341514873654, 17.3600268411254668790429899870, 18.12190622377042646440436922717, 18.87052889274706014504676239640

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