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Properties

Label 2-29e2-29.13-c1-0-51
Degree $2$
Conductor $841$
Sign $-0.553 - 0.832i$
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  + (0.702 − 1.45i)2-s + (−1.26 − 1.00i)3-s + (−0.385 − 0.483i)4-s + (−2.57 − 1.23i)5-s + (−2.35 + 1.13i)6-s + (1.39 − 1.74i)7-s + (2.18 − 0.497i)8-s + (−0.0849 − 0.372i)9-s + (−3.61 + 2.87i)10-s + (−3.52 − 0.805i)11-s + 1.00i·12-s + (0.942 − 4.12i)13-s + (−1.56 − 3.25i)14-s + (2.00 + 4.16i)15-s + (1.08 − 4.73i)16-s + 6.61i·17-s + ⋯
L(s)  = 1  + (0.496 − 1.03i)2-s + (−0.730 − 0.582i)3-s + (−0.192 − 0.241i)4-s + (−1.14 − 0.553i)5-s + (−0.962 + 0.463i)6-s + (0.526 − 0.660i)7-s + (0.770 − 0.175i)8-s + (−0.0283 − 0.124i)9-s + (−1.14 + 0.910i)10-s + (−1.06 − 0.242i)11-s + 0.288i·12-s + (0.261 − 1.14i)13-s + (−0.419 − 0.871i)14-s + (0.517 + 1.07i)15-s + (0.270 − 1.18i)16-s + 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.553 - 0.832i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.553 - 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.426267 + 0.795633i\)
\(L(\frac12)\) \(\approx\) \(0.426267 + 0.795633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.702 + 1.45i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (1.26 + 1.00i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (2.57 + 1.23i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-1.39 + 1.74i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (3.52 + 0.805i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.942 + 4.12i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 6.61iT - 17T^{2} \)
19 \( 1 + (-1.44 + 1.15i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (2.91 - 1.40i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (0.473 - 0.982i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (8.48 - 1.93i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + 2.85iT - 41T^{2} \)
43 \( 1 + (1.19 + 2.49i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-6.82 - 1.55i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.80 - 0.867i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (1.26 + 1.00i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (2.33 + 10.2i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-0.339 + 1.48i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.126 - 0.262i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (4.96 - 1.13i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-4.95 - 6.21i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-3.77 + 7.84i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (-12.9 + 10.3i)T + (21.5 - 94.5i)T^{2} \)
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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

−10.27093542373065654329610693818, −8.567514438542624997808575474724, −7.82699665260115446940160752637, −7.31135124162516674478192776900, −5.89651595437069971599711176366, −4.98407613907793877128240406886, −3.97760020827178767522547788574, −3.25127565560301820916385963631, −1.56928935141808877983371639217, −0.41539061033976021464472401644, 2.33096986653157171035170938966, 3.93058604844741250675204909030, 4.87738914011370768477976360317, 5.29853529150241532320659039234, 6.35668065614838897978663760420, 7.36468956214472346083682027704, 7.78266430533235991378141946298, 8.846451309759034450840125475590, 10.14985526883008795582614869255, 10.84128443867313087930849641866

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