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Properties

Label 2-29e2-29.6-c1-0-37
Degree $2$
Conductor $841$
Sign $-0.160 + 0.987i$
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  + (0.323 + 0.258i)2-s + (0.403 + 0.0921i)3-s + (−0.406 − 1.78i)4-s + (0.623 − 0.781i)5-s + (0.106 + 0.134i)6-s + (−0.629 + 2.75i)7-s + (0.688 − 1.42i)8-s + (−2.54 − 1.22i)9-s + (0.403 − 0.0921i)10-s + (−1.04 − 2.17i)11-s − 0.757i·12-s + (1.64 − 0.793i)13-s + (−0.915 + 0.730i)14-s + (0.323 − 0.258i)15-s + (−2.70 + 1.30i)16-s − 4.82i·17-s + ⋯
L(s)  = 1  + (0.228 + 0.182i)2-s + (0.233 + 0.0532i)3-s + (−0.203 − 0.891i)4-s + (0.278 − 0.349i)5-s + (0.0436 + 0.0547i)6-s + (−0.237 + 1.04i)7-s + (0.243 − 0.505i)8-s + (−0.849 − 0.409i)9-s + (0.127 − 0.0291i)10-s + (−0.315 − 0.655i)11-s − 0.218i·12-s + (0.456 − 0.220i)13-s + (−0.244 + 0.195i)14-s + (0.0836 − 0.0666i)15-s + (−0.675 + 0.325i)16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.160 + 0.987i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.919025 - 1.08069i\)
\(L(\frac12)\) \(\approx\) \(0.919025 - 1.08069i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.323 - 0.258i)T + (0.445 + 1.94i)T^{2} \)
3 \( 1 + (-0.403 - 0.0921i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-0.623 + 0.781i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.629 - 2.75i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (1.04 + 2.17i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-1.64 + 0.793i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 4.82iT - 17T^{2} \)
19 \( 1 + (-5.84 + 1.33i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (4.77 + 5.98i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (3.18 + 2.53i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (-1.73 + 3.60i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 + (5.01 - 3.99i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (-2.27 - 4.72i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (4.66 - 5.85i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (0.807 + 0.184i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (5.09 + 2.45i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (2.85 - 1.37i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (-3.12 + 2.49i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-0.179 + 0.373i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (-0.813 - 3.56i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (-3.50 - 2.79i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-12.1 + 2.77i)T + (87.3 - 42.0i)T^{2} \)
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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

−9.702508142985990254965564881107, −9.130757128482013586028496019190, −8.602137619978127417612438402548, −7.34282819229128988685543946669, −6.01752295551351140286718093950, −5.72198715983028540984159023691, −4.88853800747722573158164881482, −3.40310745567700563374871635830, −2.35012654452020249794377393625, −0.61407032202833060556051077461, 1.87446096312287619837477679713, 3.18647894884245717766639513296, 3.82258909254661344342571250971, 4.96208288197279987311431581710, 6.10271550191224154929890295231, 7.20889622708790624952777935227, 7.87869117372316661296382641207, 8.548699530496915881548726337449, 9.746783412154509250755854374010, 10.37240825627081281153989714685

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