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Properties

Label 2-41-41.9-c5-0-12
Degree $2$
Conductor $41$
Sign $-0.570 + 0.821i$
Analytic cond. $6.57573$
Root an. cond. $2.56431$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.81i·2-s + (3.20 + 3.20i)3-s − 1.79·4-s − 42.7i·5-s + (18.6 − 18.6i)6-s + (22.3 + 22.3i)7-s − 175. i·8-s − 222. i·9-s − 248.·10-s + (177. + 177. i)11-s + (−5.76 − 5.76i)12-s + (−580. − 580. i)13-s + (129. − 129. i)14-s + (137. − 137. i)15-s − 1.07e3·16-s + (−1.11e3 + 1.11e3i)17-s + ⋯
L(s)  = 1  − 1.02i·2-s + (0.205 + 0.205i)3-s − 0.0561·4-s − 0.765i·5-s + (0.211 − 0.211i)6-s + (0.172 + 0.172i)7-s − 0.969i·8-s − 0.915i·9-s − 0.786·10-s + (0.441 + 0.441i)11-s + (−0.0115 − 0.0115i)12-s + (−0.952 − 0.952i)13-s + (0.177 − 0.177i)14-s + (0.157 − 0.157i)15-s − 1.05·16-s + (−0.933 + 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41\)
Sign: $-0.570 + 0.821i$
Analytic conductor: \(6.57573\)
Root analytic conductor: \(2.56431\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{41} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 41,\ (\ :5/2),\ -0.570 + 0.821i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.833230 - 1.59215i\)
\(L(\frac12)\) \(\approx\) \(0.833230 - 1.59215i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad41 \( 1 + (-6.31e3 - 8.71e3i)T \)
good2 \( 1 + 5.81iT - 32T^{2} \)
3 \( 1 + (-3.20 - 3.20i)T + 243iT^{2} \)
5 \( 1 + 42.7iT - 3.12e3T^{2} \)
7 \( 1 + (-22.3 - 22.3i)T + 1.68e4iT^{2} \)
11 \( 1 + (-177. - 177. i)T + 1.61e5iT^{2} \)
13 \( 1 + (580. + 580. i)T + 3.71e5iT^{2} \)
17 \( 1 + (1.11e3 - 1.11e3i)T - 1.41e6iT^{2} \)
19 \( 1 + (-909. + 909. i)T - 2.47e6iT^{2} \)
23 \( 1 - 4.02e3T + 6.43e6T^{2} \)
29 \( 1 + (-4.25e3 - 4.25e3i)T + 2.05e7iT^{2} \)
31 \( 1 - 584.T + 2.86e7T^{2} \)
37 \( 1 + 6.54e3T + 6.93e7T^{2} \)
43 \( 1 - 1.97e4iT - 1.47e8T^{2} \)
47 \( 1 + (2.64e3 - 2.64e3i)T - 2.29e8iT^{2} \)
53 \( 1 + (-1.22e4 - 1.22e4i)T + 4.18e8iT^{2} \)
59 \( 1 + 1.84e4T + 7.14e8T^{2} \)
61 \( 1 + 1.56e4iT - 8.44e8T^{2} \)
67 \( 1 + (-1.56e4 + 1.56e4i)T - 1.35e9iT^{2} \)
71 \( 1 + (-3.29e4 - 3.29e4i)T + 1.80e9iT^{2} \)
73 \( 1 - 2.90e4iT - 2.07e9T^{2} \)
79 \( 1 + (-1.28e4 - 1.28e4i)T + 3.07e9iT^{2} \)
83 \( 1 + 3.59e4T + 3.93e9T^{2} \)
89 \( 1 + (2.57e4 + 2.57e4i)T + 5.58e9iT^{2} \)
97 \( 1 + (-3.25e4 + 3.25e4i)T - 8.58e9iT^{2} \)
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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

−14.80136444743201956876020636747, −12.89289421725528118124672565608, −12.37727421978654990144079015839, −11.10874950308752895156861213934, −9.796372927448320581357900584684, −8.793739686360963224398306408655, −6.82883867521062643836294352258, −4.70183918450259298781087235938, −2.98954843611589915637331329135, −1.05110585528430306773331802898, 2.46191022863625969667635279905, 4.98715349906891567337061949053, 6.74311616040269902734775853909, 7.43304798493928515285028464485, 8.908612695979040871197391282670, 10.73034983782650880212115485415, 11.76004111380270950055797015136, 13.78617971140720102485448151887, 14.30179171351642957031195428155, 15.47869404061953271461696471505

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