Properties

Degree 2
Conductor 41
Sign $-0.520 - 0.853i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.40 + 0.457i)2-s + (−1.90 + 1.90i)3-s + (0.153 − 0.111i)4-s + (−0.455 − 0.626i)5-s + (1.80 − 3.54i)6-s + (2.12 + 4.16i)7-s + (1.57 − 2.16i)8-s − 4.22i·9-s + (0.926 + 0.673i)10-s + (−0.538 + 0.0852i)11-s + (−0.0795 + 0.502i)12-s + (−0.808 − 0.411i)13-s + (−4.89 − 4.89i)14-s + (2.05 + 0.325i)15-s + (−1.34 + 4.13i)16-s + (0.937 + 5.91i)17-s + ⋯
L(s)  = 1  + (−0.995 + 0.323i)2-s + (−1.09 + 1.09i)3-s + (0.0765 − 0.0556i)4-s + (−0.203 − 0.280i)5-s + (0.737 − 1.44i)6-s + (0.802 + 1.57i)7-s + (0.556 − 0.766i)8-s − 1.40i·9-s + (0.293 + 0.212i)10-s + (−0.162 + 0.0257i)11-s + (−0.0229 + 0.144i)12-s + (−0.224 − 0.114i)13-s + (−1.30 − 1.30i)14-s + (0.530 + 0.0840i)15-s + (−0.335 + 1.03i)16-s + (0.227 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $-0.520 - 0.853i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ -0.520 - 0.853i)$
$L(1)$  $\approx$  $0.170582 + 0.303789i$
$L(\frac12)$  $\approx$  $0.170582 + 0.303789i$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 41$,\(F_p(T)\) is a polynomial of degree 2. If $p = 41$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad41 \( 1 + (-6.21 + 1.55i)T \)
good2 \( 1 + (1.40 - 0.457i)T + (1.61 - 1.17i)T^{2} \)
3 \( 1 + (1.90 - 1.90i)T - 3iT^{2} \)
5 \( 1 + (0.455 + 0.626i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-2.12 - 4.16i)T + (-4.11 + 5.66i)T^{2} \)
11 \( 1 + (0.538 - 0.0852i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (0.808 + 0.411i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.937 - 5.91i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-3.90 + 1.98i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (0.323 + 0.995i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.193 + 1.21i)T + (-27.5 - 8.96i)T^{2} \)
31 \( 1 + (-1.22 - 0.893i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-5.87 + 4.26i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (5.91 - 1.92i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (1.80 - 3.53i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (-0.837 + 5.28i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-2.74 - 8.44i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.31 - 0.428i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (5.56 + 0.881i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (1.07 - 0.170i)T + (67.5 - 21.9i)T^{2} \)
73 \( 1 + 5.61iT - 73T^{2} \)
79 \( 1 + (-8.62 + 8.62i)T - 79iT^{2} \)
83 \( 1 + 13.1T + 83T^{2} \)
89 \( 1 + (0.885 + 1.73i)T + (-52.3 + 72.0i)T^{2} \)
97 \( 1 + (1.91 + 0.303i)T + (92.2 + 29.9i)T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−16.55105300105210332969373269346, −15.78257907916129480269973431370, −14.87592010312843119606663684375, −12.58232684152045516294357853041, −11.51883728616917186983377722345, −10.31140457529984197977592826835, −9.155867511014742541665668274654, −8.103777273704450978782947954027, −5.88487121978187928859910847411, −4.58568720787821748340612848802, 1.06504499463299459179779644215, 5.06371086932244509324289551117, 7.15315027103536333678825845513, 7.79125557000754476571306126392, 9.873876619141543982769833970438, 11.07748584736420599724615087112, 11.64997475057130492898825374140, 13.39601410666202511683757202766, 14.26120891315344267369419804587, 16.52550290175906850670904491295

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