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Properties

Label 1-41-41.14-r1-0-0
Degree $1$
Conductor $41$
Sign $0.0750 - 0.997i$
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  + i·2-s + (−0.707 + 0.707i)3-s − 4-s − i·5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s − i·8-s − i·9-s + 10-s + (0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)14-s + (0.707 + 0.707i)15-s + 16-s + (−0.707 − 0.707i)17-s + ⋯
L(s)  = 1  + i·2-s + (−0.707 + 0.707i)3-s − 4-s − i·5-s + (−0.707 − 0.707i)6-s + (−0.707 + 0.707i)7-s − i·8-s − i·9-s + 10-s + (0.707 − 0.707i)11-s + (0.707 − 0.707i)12-s + (−0.707 + 0.707i)13-s + (−0.707 − 0.707i)14-s + (0.707 + 0.707i)15-s + 16-s + (−0.707 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0750 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0750 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(41\)
Sign: $0.0750 - 0.997i$
Analytic conductor: \(4.40606\)
Root analytic conductor: \(4.40606\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 41,\ (1:\ ),\ 0.0750 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.09395277458 - 0.08714719745i\)
\(L(\frac12)\) \(\approx\) \(0.09395277458 - 0.08714719745i\)
\(L(1)\) \(\approx\) \(0.4739978863 + 0.2394774269i\)
\(L(1)\) \(\approx\) \(0.4739978863 + 0.2394774269i\)

Euler product

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

−35.25626539498504423458854651876, −33.84032236871474399933911183494, −32.49737695645042055895998251788, −30.84605377551410170513340093805, −29.96131056439772395835896267596, −29.39632484702879269400601137994, −28.0731710716963189718022623001, −26.894340073993925138166472931383, −25.54478715866147549208346868513, −23.65320241157785521100136790211, −22.59106695930226519820226732083, −22.09426271925881593268959272572, −19.99717153676378825066373277773, −19.18931125850204004357254870598, −17.9381315226362688458747659629, −17.01750906974896711592499764038, −14.67983652534658696874753809876, −13.27163137496520295514919293084, −12.21603576801177467501379759982, −10.85143185278566022490279681615, −9.93559418919097195280507491578, −7.59871391270992737450569187906, −6.15126891806959372974453490573, −3.99357600855204847494378128355, −2.097425253162804008824615751238, 0.08542869317513882108231456845, 4.13736276530136041190886177490, 5.40016439625234311682079426459, 6.59981178430745980042527556488, 8.8616148962070003464452704085, 9.52782823977403823828418359779, 11.75287133707356890447172117554, 13.06961312221571186073633317434, 14.82056026185019933683452993693, 16.19038985902923810996035415673, 16.61277469240706781053459800711, 17.98251838472865663579967117498, 19.64554667342186736974371034334, 21.62484222974837999552395700611, 22.253795714957897503343132936426, 23.7663102726138546898729660335, 24.63682639375904715474058414749, 26.02678574161552258090082958147, 27.28763717871179903822098605444, 28.16924411126427134359552885275, 29.249349961012518161784974315059, 31.61933008831481666931620195937, 32.17419639506517574962263794686, 33.16330147620147702275437106894, 34.39810981462501564026646039873

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