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Properties

Label 1-41-41.31-r0-0-0
Degree $1$
Conductor $41$
Sign $-0.846 + 0.532i$
Analytic cond. $0.190403$
Root an. cond. $0.190403$
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  + (0.309 + 0.951i)2-s − 3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (−0.309 − 0.951i)6-s + (−0.309 + 0.951i)7-s + (−0.809 − 0.587i)8-s + 9-s + (−0.809 − 0.587i)10-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)12-s + (−0.309 − 0.951i)13-s − 14-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s − 3-s + (−0.809 + 0.587i)4-s + (−0.809 + 0.587i)5-s + (−0.309 − 0.951i)6-s + (−0.309 + 0.951i)7-s + (−0.809 − 0.587i)8-s + 9-s + (−0.809 − 0.587i)10-s + (0.809 + 0.587i)11-s + (0.809 − 0.587i)12-s + (−0.309 − 0.951i)13-s − 14-s + (0.809 − 0.587i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.846 + 0.532i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.846 + 0.532i$
Analytic conductor: \(0.190403\)
Root analytic conductor: \(0.190403\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (0:\ ),\ -0.846 + 0.532i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1492430308 + 0.5175553907i\)
\(L(\frac12)\) \(\approx\) \(0.1492430308 + 0.5175553907i\)
\(L(1)\) \(\approx\) \(0.4925508504 + 0.5018107917i\)
\(L(1)\) \(\approx\) \(0.4925508504 + 0.5018107917i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 - T \)
5 \( 1 + (-0.809 + 0.587i)T \)
7 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.809 + 0.587i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.809 - 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (0.309 + 0.951i)T \)
61 \( 1 + (0.309 - 0.951i)T \)
67 \( 1 + (0.809 - 0.587i)T \)
71 \( 1 + (0.809 + 0.587i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + T \)
89 \( 1 + (-0.309 + 0.951i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−34.579988301819513418450694125635, −32.93354317975744930136421852514, −32.186053225210674055962564290607, −30.681441559370475916618563168115, −29.60920724844508922816316743315, −28.71086370099980365408923977845, −27.54650445317085933592787259052, −26.81443431545042628047092028171, −24.20149777484618459933544545087, −23.46126371196661577720220503325, −22.45303053050163155201222148908, −21.202756978922654560064733993446, −19.832022605643103401899352306931, −18.886112689517761674164336043580, −17.17732182728307043243253111922, −16.18356778927145230233007926437, −14.17067197507874922065699052291, −12.68478519504892547967103884269, −11.70714838596729571711008167414, −10.67619335466897848766329749345, −9.10636804422757738465979246281, −6.87296095144648215520739462235, −4.922900237129970613369051488623, −3.829159915939117248735575034298, −0.895285003130252809691849780406, 3.728389671943052335117106151335, 5.428643188427879327911656212998, 6.6126218339088894340593932018, 7.956166322748873525917434154650, 9.910781191401462220192592028359, 11.84438605374654928908019785811, 12.64446239174013764142440197909, 14.80081391847283329663926805592, 15.55889752359187555467634210489, 16.85621800476937916692725043932, 18.048007944109666442587451581545, 19.168851574269370231269454928887, 21.56499284171141691879458785949, 22.62916641219154336698009424762, 23.17377969578102164707080474852, 24.59243399830760481208587897583, 25.687128569724949837384213119295, 27.39868538986403984063813108360, 27.7888413455255149144083868559, 29.72441003481582243626455971013, 30.86617531082372341906902423236, 32.09445396725929743781847026742, 33.28051358516054431481939798702, 34.51841473724129879617715349037, 34.92177190753708521901884155314

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