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Properties

Label 2-29e2-29.28-c1-0-9
Degree $2$
Conductor $841$
Sign $0.928 - 0.371i$
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  − 2.41i·2-s + 2.41i·3-s − 3.82·4-s + 5-s + 5.82·6-s − 2.82·7-s + 4.41i·8-s − 2.82·9-s − 2.41i·10-s − 0.414i·11-s − 9.24i·12-s + 3.82·13-s + 6.82i·14-s + 2.41i·15-s + 2.99·16-s + 0.828i·17-s + ⋯
L(s)  = 1  − 1.70i·2-s + 1.39i·3-s − 1.91·4-s + 0.447·5-s + 2.37·6-s − 1.06·7-s + 1.56i·8-s − 0.942·9-s − 0.763i·10-s − 0.124i·11-s − 2.66i·12-s + 1.06·13-s + 1.82i·14-s + 0.623i·15-s + 0.749·16-s + 0.200i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.928 - 0.371i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (840, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.928 - 0.371i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08265 + 0.208500i\)
\(L(\frac12)\) \(\approx\) \(1.08265 + 0.208500i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.41iT - 2T^{2} \)
3 \( 1 - 2.41iT - 3T^{2} \)
5 \( 1 - T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 0.414iT - 11T^{2} \)
13 \( 1 - 3.82T + 13T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 3.65T + 23T^{2} \)
31 \( 1 - 10.0iT - 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 - 4.48iT - 41T^{2} \)
43 \( 1 - 3.58iT - 43T^{2} \)
47 \( 1 - 3.24iT - 47T^{2} \)
53 \( 1 - 9.48T + 53T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + 4.82iT - 61T^{2} \)
67 \( 1 + 5.65T + 67T^{2} \)
71 \( 1 - 8.82T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 2.41iT - 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 + 12.4iT - 89T^{2} \)
97 \( 1 + 4.48iT - 97T^{2} \)
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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

−10.38283483988114289082580956807, −9.670914726480603372946941860759, −9.180906617620865486956159451602, −8.323379215664938491705108082300, −6.49320206463116236873727458648, −5.49506598388853544238877724714, −4.42006798095713443909397669926, −3.52542532797285951597610386244, −3.10264437230421786189216456328, −1.48273327199274240323310656581, 0.57153214305092546042254282853, 2.40105092694515958317126692962, 3.98926573870603428691897100429, 5.45111779888179061077947235226, 6.10235573169927058960045841122, 6.77608095644138810887472453383, 7.25445764760947490713352480536, 8.150111321493409223577256455376, 9.024851403525416637974269613077, 9.620358094880221700641510156493

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