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Properties

Label 1-1441-1441.458-r1-0-0
Degree $1$
Conductor $1441$
Sign $0.838 - 0.545i$
Analytic cond. $154.856$
Root an. cond. $154.856$
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  + (0.354 + 0.935i)2-s + (0.958 + 0.285i)3-s + (−0.748 + 0.663i)4-s + (−0.943 + 0.331i)5-s + (0.0724 + 0.997i)6-s + (0.527 + 0.849i)7-s + (−0.885 − 0.464i)8-s + (0.836 + 0.548i)9-s + (−0.644 − 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.926 − 0.377i)13-s + (−0.607 + 0.794i)14-s + (−0.998 + 0.0483i)15-s + (0.120 − 0.992i)16-s + (−0.981 + 0.192i)17-s + (−0.215 + 0.976i)18-s + ⋯
L(s)  = 1  + (0.354 + 0.935i)2-s + (0.958 + 0.285i)3-s + (−0.748 + 0.663i)4-s + (−0.943 + 0.331i)5-s + (0.0724 + 0.997i)6-s + (0.527 + 0.849i)7-s + (−0.885 − 0.464i)8-s + (0.836 + 0.548i)9-s + (−0.644 − 0.764i)10-s + (−0.906 + 0.421i)12-s + (−0.926 − 0.377i)13-s + (−0.607 + 0.794i)14-s + (−0.998 + 0.0483i)15-s + (0.120 − 0.992i)16-s + (−0.981 + 0.192i)17-s + (−0.215 + 0.976i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.838 - 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1441\)    =    \(11 \cdot 131\)
Sign: $0.838 - 0.545i$
Analytic conductor: \(154.856\)
Root analytic conductor: \(154.856\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1441} (458, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1441,\ (1:\ ),\ 0.838 - 0.545i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2335452288 + 0.06928231295i\)
\(L(\frac12)\) \(\approx\) \(-0.2335452288 + 0.06928231295i\)
\(L(1)\) \(\approx\) \(0.7220668384 + 0.8551235516i\)
\(L(1)\) \(\approx\) \(0.7220668384 + 0.8551235516i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
131 \( 1 \)
good2 \( 1 + (0.354 + 0.935i)T \)
3 \( 1 + (0.958 + 0.285i)T \)
5 \( 1 + (-0.943 + 0.331i)T \)
7 \( 1 + (0.527 + 0.849i)T \)
13 \( 1 + (-0.926 - 0.377i)T \)
17 \( 1 + (-0.981 + 0.192i)T \)
19 \( 1 + (-0.485 + 0.873i)T \)
23 \( 1 + (-0.443 + 0.896i)T \)
29 \( 1 + (0.998 + 0.0483i)T \)
31 \( 1 + (-0.989 + 0.144i)T \)
37 \( 1 + (-0.861 - 0.506i)T \)
41 \( 1 + (0.681 + 0.732i)T \)
43 \( 1 + (-0.958 - 0.285i)T \)
47 \( 1 + (-0.262 + 0.964i)T \)
53 \( 1 + T \)
59 \( 1 + (0.120 - 0.992i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.958 - 0.285i)T \)
71 \( 1 + (0.715 - 0.698i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.958 - 0.285i)T \)
83 \( 1 + (-0.215 - 0.976i)T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (0.644 - 0.764i)T \)
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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

−19.81973100799356872615012416862, −19.588440804637436958574523741358, −18.62345681440083562286328793117, −17.867801993764470573450322280053, −16.96662395027587450604074713410, −15.82573188327909120280517366898, −14.97645642972299122802002440981, −14.46724358619053957645527303785, −13.63361540101418427529549191518, −13.019602886513328793699382401803, −12.20383863103735003416252202199, −11.53916266201419987999102185113, −10.6476969189550712896211834920, −9.90323134072536808167741316718, −8.75845069143510282746868827062, −8.50360158618442630058538506665, −7.311095223271353805604791953831, −6.74661649085664873703074927803, −5.00822980431609982494843693383, −4.35709016170172862524807252537, −3.87296395568026373832393863562, −2.74397473482650161363171365305, −2.009625341118895592979060204674, −0.89947130717701608393734964375, −0.039875249096186543448389845826, 1.92977396747889717120030583920, 2.9245170679206715881125498860, 3.7253384778335933555377814029, 4.54010899375680765430268322805, 5.24046715354034634373183867734, 6.414441677823774173348939213274, 7.35874256500236083584135489622, 7.97532423936446174194879462530, 8.52969668602729226554670956827, 9.29131322562581190586161747894, 10.27759812416497868087340521776, 11.38874021125132439653937393508, 12.34535860298814811088464571715, 12.84778438035495272903746231779, 14.04287772502658257891054539660, 14.56519094012353462813140943204, 15.17818526757760388981447144991, 15.6585203657667968335712554469, 16.32353886127352717111825520221, 17.47299668783144110751766181100, 18.18635249953777646767827726411, 18.97779583474533923618889437144, 19.65995437743290283312021713183, 20.452779218001775129865816688942, 21.63660259986659593629921229413

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