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Properties

Label 2-29e2-29.28-c1-0-3
Degree $2$
Conductor $841$
Sign $0.814 + 0.580i$
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  + 2.26i·2-s + 2.84i·3-s − 3.14·4-s − 0.0578·5-s − 6.44·6-s − 1.56·7-s − 2.59i·8-s − 5.06·9-s − 0.131i·10-s − 3.97i·11-s − 8.92i·12-s − 0.404·13-s − 3.54i·14-s − 0.164i·15-s − 0.408·16-s + 5.16i·17-s + ⋯
L(s)  = 1  + 1.60i·2-s + 1.64i·3-s − 1.57·4-s − 0.0258·5-s − 2.62·6-s − 0.590·7-s − 0.916i·8-s − 1.68·9-s − 0.0415i·10-s − 1.19i·11-s − 2.57i·12-s − 0.112·13-s − 0.946i·14-s − 0.0424i·15-s − 0.102·16-s + 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 + 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.814 + 0.580i$
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: \((2,\ 841,\ (\ :1/2),\ 0.814 + 0.580i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.520476 - 0.166615i\)
\(L(\frac12)\) \(\approx\) \(0.520476 - 0.166615i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.26iT - 2T^{2} \)
3 \( 1 - 2.84iT - 3T^{2} \)
5 \( 1 + 0.0578T + 5T^{2} \)
7 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 + 3.97iT - 11T^{2} \)
13 \( 1 + 0.404T + 13T^{2} \)
17 \( 1 - 5.16iT - 17T^{2} \)
19 \( 1 + 3.45iT - 19T^{2} \)
23 \( 1 - 0.230T + 23T^{2} \)
31 \( 1 - 4.31iT - 31T^{2} \)
37 \( 1 + 5.37iT - 37T^{2} \)
41 \( 1 - 1.46iT - 41T^{2} \)
43 \( 1 - 6.48iT - 43T^{2} \)
47 \( 1 - 9.57iT - 47T^{2} \)
53 \( 1 + 1.78T + 53T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 + 3.18iT - 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 + 13.4T + 71T^{2} \)
73 \( 1 - 7.01iT - 73T^{2} \)
79 \( 1 - 4.78iT - 79T^{2} \)
83 \( 1 - 8.42T + 83T^{2} \)
89 \( 1 + 1.13iT - 89T^{2} \)
97 \( 1 - 17.2iT - 97T^{2} \)
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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

−10.75836907020151247631595132335, −9.824213365303280992889089405015, −9.083651272890841930960928875100, −8.527909010838303366240815183562, −7.62500017594790809390817461154, −6.30595784002917407684798170908, −5.87200818524451700304835908037, −4.90658670364304278303894427330, −4.06065859109562486050059741770, −3.11681521109527789878877365021, 0.25992710056694788157743513441, 1.65194278459982401091304052878, 2.36718407704002171412828214218, 3.40241343153496447653524117749, 4.68095237738504486205538091707, 6.00537525713289421407408640962, 7.03928726531916786560137677792, 7.62318646238388420955288395041, 8.791176338107895647459649332919, 9.711548833936985742591282832147

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