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Properties

Label 1-41-41.35-r1-0-0
Degree $1$
Conductor $41$
Sign $-0.843 + 0.537i$
Analytic cond. $4.40606$
Root an. cond. $4.40606$
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  + (−0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (−0.951 − 0.309i)5-s + (−0.987 − 0.156i)6-s + (−0.987 + 0.156i)7-s + (0.951 − 0.309i)8-s − i·9-s + (0.309 + 0.951i)10-s + (−0.891 − 0.453i)11-s + (0.453 + 0.891i)12-s + (−0.156 + 0.987i)13-s + (0.707 + 0.707i)14-s + (−0.891 + 0.453i)15-s + (−0.809 − 0.587i)16-s + (−0.453 + 0.891i)17-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (0.707 − 0.707i)3-s + (−0.309 + 0.951i)4-s + (−0.951 − 0.309i)5-s + (−0.987 − 0.156i)6-s + (−0.987 + 0.156i)7-s + (0.951 − 0.309i)8-s − i·9-s + (0.309 + 0.951i)10-s + (−0.891 − 0.453i)11-s + (0.453 + 0.891i)12-s + (−0.156 + 0.987i)13-s + (0.707 + 0.707i)14-s + (−0.891 + 0.453i)15-s + (−0.809 − 0.587i)16-s + (−0.453 + 0.891i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.843 + 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $-0.843 + 0.537i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ -0.843 + 0.537i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1219312920 - 0.4178552160i\)
\(L(\frac12)\) \(\approx\) \(-0.1219312920 - 0.4178552160i\)
\(L(1)\) \(\approx\) \(0.4436780800 - 0.4009362910i\)
\(L(1)\) \(\approx\) \(0.4436780800 - 0.4009362910i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 \)
good2 \( 1 + (-0.587 - 0.809i)T \)
3 \( 1 + (0.707 - 0.707i)T \)
5 \( 1 + (-0.951 - 0.309i)T \)
7 \( 1 + (-0.987 + 0.156i)T \)
11 \( 1 + (-0.891 - 0.453i)T \)
13 \( 1 + (-0.156 + 0.987i)T \)
17 \( 1 + (-0.453 + 0.891i)T \)
19 \( 1 + (-0.156 - 0.987i)T \)
23 \( 1 + (0.809 - 0.587i)T \)
29 \( 1 + (-0.453 - 0.891i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.587 - 0.809i)T \)
47 \( 1 + (-0.987 - 0.156i)T \)
53 \( 1 + (0.453 + 0.891i)T \)
59 \( 1 + (-0.809 + 0.587i)T \)
61 \( 1 + (0.587 - 0.809i)T \)
67 \( 1 + (0.891 - 0.453i)T \)
71 \( 1 + (0.891 + 0.453i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.987 + 0.156i)T \)
97 \( 1 + (-0.891 + 0.453i)T \)
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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

−35.406464564821670214858798431543, −34.1515344814477609716112608959, −33.061297550159361463909242657345, −31.91659049226436395498116774110, −31.1940386277135987729830534233, −29.17725840781471748257536724029, −27.67962623124966842394376939521, −26.934807592616775482069421960459, −25.92539815870658932707496321247, −25.029816001442385631528986902337, −23.293384657765226339493267257476, −22.484523219304329762053687016496, −20.3644300843454339832548743317, −19.50504117845041949781200446218, −18.32963977090478960401120195788, −16.42038983523539110685971545404, −15.64179510853793667768285502484, −14.754153053438757591589535073252, −13.12243901834824771312728433075, −10.7180341930161342724277016665, −9.69294072519639419169968340872, −8.21322079413197936365929003854, −7.13382404571849735863605510848, −5.01594336548300169329515616979, −3.19950903700724516860825017497, 0.293532809634490773928706235576, 2.50989426397282004739210628317, 3.92349027333154157023698506619, 6.97283483782186381511856246750, 8.30713561567245405467197362873, 9.31823639179916689025054978198, 11.191817851997362377627940504492, 12.5692596413842919198890408649, 13.338412325874074498012823896, 15.42483345634959340782516027988, 16.83508781948868701368519359784, 18.641609962778824395701553121694, 19.24682565041557751171234651163, 20.1605235055494734008123874095, 21.5047886013485078255870753193, 23.17129473631257978585631953432, 24.42062730162095619439752861984, 26.11349627732762239487832314497, 26.536062580776336151174579627766, 28.33422156252420842532061743533, 29.108712065001454156805144345129, 30.47368025628943931443543000512, 31.38330886740571678451778139691, 32.151916305722126442806008156511, 34.6547379321028542429700857192

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