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Properties

Label 2-29e2-29.5-c1-0-38
Degree $2$
Conductor $841$
Sign $0.959 + 0.280i$
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  + (1.26 − 1.00i)2-s + (1.57 − 0.360i)3-s + (0.137 − 0.602i)4-s + (1.77 + 2.23i)5-s + (1.63 − 2.04i)6-s + (−0.497 − 2.18i)7-s + (0.970 + 2.01i)8-s + (−0.344 + 0.165i)9-s + (4.50 + 1.02i)10-s + (−1.56 + 3.25i)11-s − 1.00i·12-s + (3.81 + 1.83i)13-s + (−2.82 − 2.25i)14-s + (3.61 + 2.87i)15-s + (4.37 + 2.10i)16-s − 6.61i·17-s + ⋯
L(s)  = 1  + (0.894 − 0.713i)2-s + (0.910 − 0.207i)3-s + (0.0687 − 0.301i)4-s + (0.795 + 0.997i)5-s + (0.666 − 0.835i)6-s + (−0.188 − 0.823i)7-s + (0.343 + 0.712i)8-s + (−0.114 + 0.0552i)9-s + (1.42 + 0.324i)10-s + (−0.473 + 0.982i)11-s − 0.288i·12-s + (1.05 + 0.509i)13-s + (−0.755 − 0.602i)14-s + (0.932 + 0.743i)15-s + (1.09 + 0.526i)16-s − 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.56357 - 0.510476i\)
\(L(\frac12)\) \(\approx\) \(3.56357 - 0.510476i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.26 + 1.00i)T + (0.445 - 1.94i)T^{2} \)
3 \( 1 + (-1.57 + 0.360i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (-1.77 - 2.23i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.497 + 2.18i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (1.56 - 3.25i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-3.81 - 1.83i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 + 6.61iT - 17T^{2} \)
19 \( 1 + (1.80 + 0.412i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-2.01 + 2.53i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (0.852 - 0.679i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (3.77 + 7.84i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 2.85iT - 41T^{2} \)
43 \( 1 + (2.16 + 1.72i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-3.03 + 6.30i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.24 + 1.56i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (-1.57 + 0.360i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (9.43 - 4.54i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-1.37 - 0.662i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.228 - 0.181i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (2.20 + 4.58i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (1.76 - 7.74i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-6.80 + 5.42i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (16.1 + 3.68i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.43799395037140158032300775303, −9.441775439072635246256442818914, −8.533374657031389249073099125875, −7.40104474000624904890102273759, −6.87023441184350434624557932110, −5.58441961295847694734647593917, −4.49122125106732550969704939503, −3.47183867962492087864633506577, −2.66562146827826425217981085698, −1.95645642266319388705832797008, 1.46872239749463051132675734107, 3.03333062305319084686000263179, 3.89145938384438975128674028620, 5.13197325992148277534690842822, 5.94472390643231611120454723163, 6.15259605036803085048843365473, 7.903961569297964390331995437891, 8.618188044414003652862840199657, 9.052160203208064210463738058588, 10.06682987919344548204969540805

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