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Properties

Label 2-29e2-29.28-c1-0-22
Degree $2$
Conductor $841$
Sign $0.689 - 0.724i$
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  + 2.60i·2-s + 0.439i·3-s − 4.77·4-s − 2.58·5-s − 1.14·6-s + 0.0751·7-s − 7.21i·8-s + 2.80·9-s − 6.71i·10-s − 3.77i·11-s − 2.09i·12-s − 0.880·13-s + 0.195i·14-s − 1.13i·15-s + 9.23·16-s − 3.94i·17-s + ⋯
L(s)  = 1  + 1.84i·2-s + 0.253i·3-s − 2.38·4-s − 1.15·5-s − 0.466·6-s + 0.0283·7-s − 2.55i·8-s + 0.935·9-s − 2.12i·10-s − 1.13i·11-s − 0.605i·12-s − 0.244·13-s + 0.0522i·14-s − 0.292i·15-s + 2.30·16-s − 0.955i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 - 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.689 - 0.724i$
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: \((2,\ 841,\ (\ :1/2),\ 0.689 - 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.718040 + 0.307744i\)
\(L(\frac12)\) \(\approx\) \(0.718040 + 0.307744i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.60iT - 2T^{2} \)
3 \( 1 - 0.439iT - 3T^{2} \)
5 \( 1 + 2.58T + 5T^{2} \)
7 \( 1 - 0.0751T + 7T^{2} \)
11 \( 1 + 3.77iT - 11T^{2} \)
13 \( 1 + 0.880T + 13T^{2} \)
17 \( 1 + 3.94iT - 17T^{2} \)
19 \( 1 + 0.713iT - 19T^{2} \)
23 \( 1 - 1.17T + 23T^{2} \)
31 \( 1 + 5.15iT - 31T^{2} \)
37 \( 1 + 3.08iT - 37T^{2} \)
41 \( 1 - 6.67iT - 41T^{2} \)
43 \( 1 - 8.31iT - 43T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 - 5.55T + 53T^{2} \)
59 \( 1 - 9.91T + 59T^{2} \)
61 \( 1 + 3.56iT - 61T^{2} \)
67 \( 1 + 4.93T + 67T^{2} \)
71 \( 1 + 4.90T + 71T^{2} \)
73 \( 1 + 8.90iT - 73T^{2} \)
79 \( 1 + 13.1iT - 79T^{2} \)
83 \( 1 - 16.9T + 83T^{2} \)
89 \( 1 + 6.79iT - 89T^{2} \)
97 \( 1 + 11.4iT - 97T^{2} \)
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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

−9.920966214398783080273208209804, −9.153097088009776683707825664871, −8.293577523028256560441624922299, −7.64322960497078491328310676337, −7.02893706398074459103685689744, −6.10530017466064690141980970914, −5.03764070663355055301094128186, −4.33377021365207917069497963352, −3.40648220553865897235261908723, −0.45103988782403330640707664282, 1.25599243196114125547833997660, 2.32234474451306344188214820178, 3.69220174697756410179255644958, 4.20466583423169203956532816245, 5.09471988890565424378810029778, 6.87506652278963939948084461881, 7.75793205393886450943802834099, 8.609639493956119246710766572641, 9.594468543633906389944746470608, 10.30100049597531943476305756540

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