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Properties

Label 2-29e2-29.28-c1-0-24
Degree $2$
Conductor $841$
Sign $0.648 + 0.760i$
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  − 0.445i·2-s − 1.24i·3-s + 1.80·4-s − 0.692·5-s − 0.554·6-s + 0.356·7-s − 1.69i·8-s + 1.44·9-s + 0.307i·10-s + 4.93i·11-s − 2.24i·12-s + 5.65·13-s − 0.158i·14-s + 0.862i·15-s + 2.85·16-s + 4.49i·17-s + ⋯
L(s)  = 1  − 0.314i·2-s − 0.719i·3-s + 0.900·4-s − 0.309·5-s − 0.226·6-s + 0.134·7-s − 0.598i·8-s + 0.481·9-s + 0.0973i·10-s + 1.48i·11-s − 0.648i·12-s + 1.56·13-s − 0.0424i·14-s + 0.222i·15-s + 0.712·16-s + 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.648 + 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.648 + 0.760i$
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: \((2,\ 841,\ (\ :1/2),\ 0.648 + 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90538 - 0.879359i\)
\(L(\frac12)\) \(\approx\) \(1.90538 - 0.879359i\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.445iT - 2T^{2} \)
3 \( 1 + 1.24iT - 3T^{2} \)
5 \( 1 + 0.692T + 5T^{2} \)
7 \( 1 - 0.356T + 7T^{2} \)
11 \( 1 - 4.93iT - 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 4.49iT - 17T^{2} \)
19 \( 1 + 2.35iT - 19T^{2} \)
23 \( 1 - 2.29T + 23T^{2} \)
31 \( 1 + 6.69iT - 31T^{2} \)
37 \( 1 - 4.93iT - 37T^{2} \)
41 \( 1 + 3.10iT - 41T^{2} \)
43 \( 1 + 3.40iT - 43T^{2} \)
47 \( 1 - 6.44iT - 47T^{2} \)
53 \( 1 + 4.69T + 53T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + 1.64iT - 61T^{2} \)
67 \( 1 - 2.32T + 67T^{2} \)
71 \( 1 - 7.33T + 71T^{2} \)
73 \( 1 + 5.62iT - 73T^{2} \)
79 \( 1 + 4.66iT - 79T^{2} \)
83 \( 1 - 4.45T + 83T^{2} \)
89 \( 1 + 5.67iT - 89T^{2} \)
97 \( 1 - 0.180iT - 97T^{2} \)
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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

−10.25047247904797407495608646531, −9.367451893117783110384348053978, −8.058407249742578242267916884719, −7.55154436405546818867981896091, −6.62714676522282349383370458380, −6.08161817968317707364753267775, −4.49898816108234616844708057071, −3.56251006901415576557139193584, −2.10446250264850706279372336461, −1.37131935278624560137010563847, 1.35547734050390758203242567585, 3.10878710753266095419198814711, 3.77669367836919780328807930968, 5.11703541094324348790220189404, 5.98624677993678260908880426271, 6.78844032774796280760882051293, 7.83388697064111815165454142840, 8.523886557882103057672954230619, 9.450064854752930094913419204511, 10.54622212755804094697376523685

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