Properties

Degree 2
Conductor 41
Sign $0.603 + 0.797i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.42 − 1.96i)2-s + (−2.02 + 2.02i)3-s + (−1.20 − 3.69i)4-s + (0.110 − 0.0358i)5-s + (1.08 + 6.86i)6-s + (−0.422 + 2.66i)7-s + (−4.36 − 1.41i)8-s − 5.19i·9-s + (0.0870 − 0.267i)10-s + (−1.53 − 3.01i)11-s + (9.92 + 5.05i)12-s + (−1.03 + 0.164i)13-s + (4.63 + 4.63i)14-s + (−0.150 + 0.296i)15-s + (−2.71 + 1.97i)16-s + (3.25 − 1.66i)17-s + ⋯
L(s)  = 1  + (1.00 − 1.38i)2-s + (−1.16 + 1.16i)3-s + (−0.601 − 1.84i)4-s + (0.0493 − 0.0160i)5-s + (0.443 + 2.80i)6-s + (−0.159 + 1.00i)7-s + (−1.54 − 0.501i)8-s − 1.73i·9-s + (0.0275 − 0.0847i)10-s + (−0.463 − 0.909i)11-s + (2.86 + 1.45i)12-s + (−0.288 + 0.0456i)13-s + (1.23 + 1.23i)14-s + (−0.0389 + 0.0764i)15-s + (−0.678 + 0.492i)16-s + (0.790 − 0.402i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(41\)
\( \varepsilon \)  =  $0.603 + 0.797i$
motivic weight  =  \(1\)
character  :  $\chi_{41} (33, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 41,\ (\ :1/2),\ 0.603 + 0.797i)$
$L(1)$  $\approx$  $0.789427 - 0.392574i$
$L(\frac12)$  $\approx$  $0.789427 - 0.392574i$
$L(\frac{3}{2})$   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 \( 1 + (1.32 - 6.26i)T \)
good2 \( 1 + (-1.42 + 1.96i)T + (-0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.02 - 2.02i)T - 3iT^{2} \)
5 \( 1 + (-0.110 + 0.0358i)T + (4.04 - 2.93i)T^{2} \)
7 \( 1 + (0.422 - 2.66i)T + (-6.65 - 2.16i)T^{2} \)
11 \( 1 + (1.53 + 3.01i)T + (-6.46 + 8.89i)T^{2} \)
13 \( 1 + (1.03 - 0.164i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (-3.25 + 1.66i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (2.25 + 0.356i)T + (18.0 + 5.87i)T^{2} \)
23 \( 1 + (-6.06 - 4.40i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-1.31 - 0.669i)T + (17.0 + 23.4i)T^{2} \)
31 \( 1 + (0.964 - 2.96i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.348 + 1.07i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.583 + 0.803i)T + (-13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.73 + 10.9i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-0.482 - 0.246i)T + (31.1 + 42.8i)T^{2} \)
59 \( 1 + (1.57 + 1.14i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (5.95 + 8.20i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + (6.02 - 11.8i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (4.28 + 8.40i)T + (-41.7 + 57.4i)T^{2} \)
73 \( 1 - 10.9iT - 73T^{2} \)
79 \( 1 + (-7.58 + 7.58i)T - 79iT^{2} \)
83 \( 1 + 0.635T + 83T^{2} \)
89 \( 1 + (-0.753 + 4.75i)T + (-84.6 - 27.5i)T^{2} \)
97 \( 1 + (1.25 - 2.46i)T + (-57.0 - 78.4i)T^{2} \)
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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

−15.76497094481683303422189625278, −14.81113720738231300335300805053, −13.28061946905238215423370945177, −12.03803573331724070489269623313, −11.35978643855053120847750624640, −10.39155401516290997101587820727, −9.300320859355091857203263397131, −5.75027550235750337227493102359, −5.02931207858083653330124902949, −3.30540078809108426168597042181, 4.61801837677553583998886778041, 5.99467357467961706684897622328, 7.03438274842776910036727226453, 7.78118523104727359568711328734, 10.54335536350074097252159332843, 12.27609312528662628552278594407, 12.90993492529208515572899741622, 13.88428915700876021786797291770, 15.12103069889448822146076073715, 16.56032008359819408333257257178

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