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Properties

Label 2-41-41.14-c2-0-3
Degree $2$
Conductor $41$
Sign $0.497 + 0.867i$
Analytic cond. $1.11716$
Root an. cond. $1.05696$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.976 − 0.976i)2-s + (0.355 + 0.857i)3-s − 2.09i·4-s + (4.54 − 4.54i)5-s + (0.490 − 1.18i)6-s + (0.392 + 0.947i)7-s + (−5.95 + 5.95i)8-s + (5.75 − 5.75i)9-s − 8.87·10-s + (−12.1 + 5.04i)11-s + (1.79 − 0.742i)12-s + (9.50 + 22.9i)13-s + (0.542 − 1.30i)14-s + (5.50 + 2.28i)15-s + 3.25·16-s + (0.0103 − 0.0250i)17-s + ⋯
L(s)  = 1  + (−0.488 − 0.488i)2-s + (0.118 + 0.285i)3-s − 0.522i·4-s + (0.908 − 0.908i)5-s + (0.0817 − 0.197i)6-s + (0.0560 + 0.135i)7-s + (−0.743 + 0.743i)8-s + (0.639 − 0.639i)9-s − 0.887·10-s + (−1.10 + 0.459i)11-s + (0.149 − 0.0618i)12-s + (0.730 + 1.76i)13-s + (0.0387 − 0.0935i)14-s + (0.367 + 0.152i)15-s + 0.203·16-s + (0.000610 − 0.00147i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.497 + 0.867i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41\)
Sign: $0.497 + 0.867i$
Analytic conductor: \(1.11716\)
Root analytic conductor: \(1.05696\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 41,\ (\ :1),\ 0.497 + 0.867i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.836365 - 0.484356i\)
\(L(\frac12)\) \(\approx\) \(0.836365 - 0.484356i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 + (36.9 - 17.6i)T \)
good2 \( 1 + (0.976 + 0.976i)T + 4iT^{2} \)
3 \( 1 + (-0.355 - 0.857i)T + (-6.36 + 6.36i)T^{2} \)
5 \( 1 + (-4.54 + 4.54i)T - 25iT^{2} \)
7 \( 1 + (-0.392 - 0.947i)T + (-34.6 + 34.6i)T^{2} \)
11 \( 1 + (12.1 - 5.04i)T + (85.5 - 85.5i)T^{2} \)
13 \( 1 + (-9.50 - 22.9i)T + (-119. + 119. i)T^{2} \)
17 \( 1 + (-0.0103 + 0.0250i)T + (-204. - 204. i)T^{2} \)
19 \( 1 + (4.10 - 9.91i)T + (-255. - 255. i)T^{2} \)
23 \( 1 + 26.1iT - 529T^{2} \)
29 \( 1 + (9.09 + 21.9i)T + (-594. + 594. i)T^{2} \)
31 \( 1 - 34.6iT - 961T^{2} \)
37 \( 1 - 31.8T + 1.36e3T^{2} \)
43 \( 1 + (26.6 + 26.6i)T + 1.84e3iT^{2} \)
47 \( 1 + (5.31 - 12.8i)T + (-1.56e3 - 1.56e3i)T^{2} \)
53 \( 1 + (44.5 - 18.4i)T + (1.98e3 - 1.98e3i)T^{2} \)
59 \( 1 + 2.86T + 3.48e3T^{2} \)
61 \( 1 + (29.7 + 29.7i)T + 3.72e3iT^{2} \)
67 \( 1 + (-14.9 + 36.1i)T + (-3.17e3 - 3.17e3i)T^{2} \)
71 \( 1 + (22.4 + 54.1i)T + (-3.56e3 + 3.56e3i)T^{2} \)
73 \( 1 + (56.7 + 56.7i)T + 5.32e3iT^{2} \)
79 \( 1 + (134. - 55.9i)T + (4.41e3 - 4.41e3i)T^{2} \)
83 \( 1 - 127.T + 6.88e3T^{2} \)
89 \( 1 + (39.5 + 95.5i)T + (-5.60e3 + 5.60e3i)T^{2} \)
97 \( 1 + (-109. - 45.5i)T + (6.65e3 + 6.65e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.75929700713343151479457467023, −14.50526904426518012428188058074, −13.28130353225146714002849292585, −12.03112769637899742148616665865, −10.48040583027418947059536829237, −9.547868904178876252910203332385, −8.712290001507525518803755118768, −6.31454494349761415716149135075, −4.75936157613422046500314518541, −1.76362477396277315313963438749, 2.95246390253253029243403576245, 5.79202293882790490037079232590, 7.29557544235212164443249500027, 8.213613269414153210607207742456, 9.959573468363943750427420272940, 10.93212087962461569125618212690, 13.04438758358029959875752417775, 13.42164893115747142737724602888, 15.18546055866606602440436265648, 16.05616776266590672612501800532

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