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Properties

Label 2-41-41.31-c3-0-3
Degree $2$
Conductor $41$
Sign $0.401 - 0.916i$
Analytic cond. $2.41907$
Root an. cond. $1.55533$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (2.80 + 2.03i)2-s + 1.54i·3-s + (1.24 + 3.82i)4-s + (3.68 + 11.3i)5-s + (−3.14 + 4.32i)6-s + (−3.25 − 4.48i)7-s + (4.26 − 13.1i)8-s + 24.6·9-s + (−12.7 + 39.3i)10-s + (−58.9 − 19.1i)11-s + (−5.89 + 1.91i)12-s + (24.0 − 33.1i)13-s − 19.2i·14-s + (−17.4 + 5.68i)15-s + (64.7 − 47.0i)16-s + (−28.7 − 9.33i)17-s + ⋯
L(s)  = 1  + (0.991 + 0.720i)2-s + 0.296i·3-s + (0.155 + 0.478i)4-s + (0.329 + 1.01i)5-s + (−0.213 + 0.294i)6-s + (−0.175 − 0.242i)7-s + (0.188 − 0.579i)8-s + 0.912·9-s + (−0.404 + 1.24i)10-s + (−1.61 − 0.525i)11-s + (−0.141 + 0.0461i)12-s + (0.513 − 0.706i)13-s − 0.366i·14-s + (−0.301 + 0.0978i)15-s + (1.01 − 0.734i)16-s + (−0.409 − 0.133i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.401 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41\)
Sign: $0.401 - 0.916i$
Analytic conductor: \(2.41907\)
Root analytic conductor: \(1.55533\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 41,\ (\ :3/2),\ 0.401 - 0.916i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.73400 + 1.13371i\)
\(L(\frac12)\) \(\approx\) \(1.73400 + 1.13371i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 + (-38.9 - 259. i)T \)
good2 \( 1 + (-2.80 - 2.03i)T + (2.47 + 7.60i)T^{2} \)
3 \( 1 - 1.54iT - 27T^{2} \)
5 \( 1 + (-3.68 - 11.3i)T + (-101. + 73.4i)T^{2} \)
7 \( 1 + (3.25 + 4.48i)T + (-105. + 326. i)T^{2} \)
11 \( 1 + (58.9 + 19.1i)T + (1.07e3 + 782. i)T^{2} \)
13 \( 1 + (-24.0 + 33.1i)T + (-678. - 2.08e3i)T^{2} \)
17 \( 1 + (28.7 + 9.33i)T + (3.97e3 + 2.88e3i)T^{2} \)
19 \( 1 + (-27.2 - 37.4i)T + (-2.11e3 + 6.52e3i)T^{2} \)
23 \( 1 + (38.1 + 27.6i)T + (3.75e3 + 1.15e4i)T^{2} \)
29 \( 1 + (201. - 65.5i)T + (1.97e4 - 1.43e4i)T^{2} \)
31 \( 1 + (43.5 - 133. i)T + (-2.41e4 - 1.75e4i)T^{2} \)
37 \( 1 + (-83.7 - 257. i)T + (-4.09e4 + 2.97e4i)T^{2} \)
43 \( 1 + (94.1 + 68.4i)T + (2.45e4 + 7.56e4i)T^{2} \)
47 \( 1 + (5.34 - 7.35i)T + (-3.20e4 - 9.87e4i)T^{2} \)
53 \( 1 + (44.5 - 14.4i)T + (1.20e5 - 8.75e4i)T^{2} \)
59 \( 1 + (552. + 401. i)T + (6.34e4 + 1.95e5i)T^{2} \)
61 \( 1 + (-498. + 362. i)T + (7.01e4 - 2.15e5i)T^{2} \)
67 \( 1 + (709. - 230. i)T + (2.43e5 - 1.76e5i)T^{2} \)
71 \( 1 + (-1.01e3 - 330. i)T + (2.89e5 + 2.10e5i)T^{2} \)
73 \( 1 + 368.T + 3.89e5T^{2} \)
79 \( 1 + 628. iT - 4.93e5T^{2} \)
83 \( 1 - 1.11e3T + 5.71e5T^{2} \)
89 \( 1 + (-192. - 264. i)T + (-2.17e5 + 6.70e5i)T^{2} \)
97 \( 1 + (-1.13e3 + 369. i)T + (7.38e5 - 5.36e5i)T^{2} \)
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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

−15.61972332048448104012385401125, −14.74739447360566912502846712813, −13.54413338637414705712872768836, −12.90104349968091725782063900300, −10.74683717048554819283081224435, −10.03580160674763398550708384490, −7.70656316467169353545867455238, −6.46634872098197297895023344963, −5.15317533799100525903801644212, −3.36846310469279576965261393691, 2.06981828455298793889110304523, 4.29671212576967931721574896737, 5.52219070862029154001903573073, 7.68141042015249059934593049535, 9.304812718098006760818595973936, 10.84300251335676011332334531913, 12.26056595427648018671165337765, 13.07145624957767399330350800072, 13.52026985715765234251038520844, 15.28630681978890736340632313461

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