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Properties

Label 2-29e2-29.23-c1-0-49
Degree $2$
Conductor $841$
Sign $-0.872 + 0.488i$
Analytic cond. $6.71541$
Root an. cond. $2.59141$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.62 − 2.03i)2-s + (−0.0977 + 0.428i)3-s + (−1.06 − 4.65i)4-s + (1.60 − 2.01i)5-s + (0.712 + 0.893i)6-s + (−0.0167 + 0.0732i)7-s + (−6.50 − 3.13i)8-s + (2.52 + 1.21i)9-s + (−1.49 − 6.54i)10-s + (3.40 − 1.63i)11-s + 2.09·12-s + (−0.793 + 0.382i)13-s + (0.121 + 0.152i)14-s + (0.706 + 0.886i)15-s + (−8.31 + 4.00i)16-s − 3.94·17-s + ⋯
L(s)  = 1  + (1.14 − 1.43i)2-s + (−0.0564 + 0.247i)3-s + (−0.531 − 2.32i)4-s + (0.719 − 0.902i)5-s + (0.291 + 0.364i)6-s + (−0.00631 + 0.0276i)7-s + (−2.29 − 1.10i)8-s + (0.843 + 0.405i)9-s + (−0.472 − 2.07i)10-s + (1.02 − 0.494i)11-s + 0.605·12-s + (−0.220 + 0.106i)13-s + (0.0325 + 0.0408i)14-s + (0.182 + 0.228i)15-s + (−2.07 + 1.00i)16-s − 0.955·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.872 + 0.488i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.872 + 0.488i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.872 + 0.488i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.794939 - 3.04794i\)
\(L(\frac12)\) \(\approx\) \(0.794939 - 3.04794i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 \)
good2 \( 1 + (-1.62 + 2.03i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (0.0977 - 0.428i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (-1.60 + 2.01i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.0167 - 0.0732i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-3.40 + 1.63i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.793 - 0.382i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 3.94T + 17T^{2} \)
19 \( 1 + (-0.158 - 0.695i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-0.734 - 0.920i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (3.21 - 4.02i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (2.78 + 1.33i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 6.67T + 41T^{2} \)
43 \( 1 + (-5.18 - 6.49i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (9.54 - 4.59i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (-3.46 + 4.34i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 9.91T + 59T^{2} \)
61 \( 1 + (-0.793 + 3.47i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (4.45 + 2.14i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (4.42 - 2.12i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-5.55 - 6.95i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-11.8 - 5.72i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (3.77 + 16.5i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (4.23 - 5.31i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (2.54 + 11.1i)T + (-87.3 + 42.0i)T^{2} \)
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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

−9.966708239973770529543732893587, −9.407582995442399370015327084041, −8.649169791861552420352069176019, −6.91160119625686386264912750675, −5.86917652230221585101712142502, −5.01901893181117017946878154321, −4.38660227800980275863902855251, −3.45896244649481795146075173314, −2.03285672921803329222279713706, −1.24354751595597887009114867788, 2.19802258653060331951596785526, 3.60940403947692365587761933701, 4.38538579976201922050162935670, 5.46089176831185766102553242806, 6.50028239682657027171213293471, 6.78153124570484683426462655388, 7.41685675529017624672913137347, 8.647706239775866473159597079382, 9.529260173127258145369650498539, 10.50208461217302009654280126196

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