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Properties

Label 2-29e2-29.20-c1-0-46
Degree $2$
Conductor $841$
Sign $0.897 + 0.440i$
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  + (2.17 − 1.04i)2-s + (1.50 + 1.88i)3-s + (2.38 − 2.99i)4-s + (0.900 − 0.433i)5-s + (5.25 + 2.52i)6-s + (−1.76 − 2.21i)7-s + (0.982 − 4.30i)8-s + (−0.629 + 2.75i)9-s + (1.50 − 1.88i)10-s + (0.0921 + 0.403i)11-s + 9.24·12-s + (0.851 + 3.73i)13-s + (−6.15 − 2.96i)14-s + (2.17 + 1.04i)15-s + (−0.667 − 2.92i)16-s + 0.828·17-s + ⋯
L(s)  = 1  + (1.53 − 0.740i)2-s + (0.869 + 1.08i)3-s + (1.19 − 1.49i)4-s + (0.402 − 0.194i)5-s + (2.14 + 1.03i)6-s + (−0.666 − 0.835i)7-s + (0.347 − 1.52i)8-s + (−0.209 + 0.919i)9-s + (0.475 − 0.596i)10-s + (0.0277 + 0.121i)11-s + 2.66·12-s + (0.236 + 1.03i)13-s + (−1.64 − 0.791i)14-s + (0.561 + 0.270i)15-s + (−0.166 − 0.731i)16-s + 0.200·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.897 + 0.440i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.897 + 0.440i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.62519 - 1.07457i\)
\(L(\frac12)\) \(\approx\) \(4.62519 - 1.07457i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.17 + 1.04i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (-1.50 - 1.88i)T + (-0.667 + 2.92i)T^{2} \)
5 \( 1 + (-0.900 + 0.433i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (1.76 + 2.21i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (-0.0921 - 0.403i)T + (-9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.851 - 3.73i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 - 0.828T + 17T^{2} \)
19 \( 1 + (-3.74 + 4.69i)T + (-4.22 - 18.5i)T^{2} \)
23 \( 1 + (3.29 + 1.58i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (9.07 - 4.36i)T + (19.3 - 24.2i)T^{2} \)
37 \( 1 + (-0.890 + 3.89i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 4.48T + 41T^{2} \)
43 \( 1 + (3.23 + 1.55i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (-0.721 - 3.16i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (8.54 - 4.11i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + (3.01 + 3.77i)T + (-13.5 + 59.4i)T^{2} \)
67 \( 1 + (1.25 - 5.51i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (-1.96 - 8.60i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (3.60 + 1.73i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-0.537 + 2.35i)T + (-71.1 - 34.2i)T^{2} \)
83 \( 1 + (-4.77 + 5.98i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (-11.2 + 5.41i)T + (55.4 - 69.5i)T^{2} \)
97 \( 1 + (-2.79 + 3.50i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.23928543007227661426759539969, −9.516375633684627774869056152274, −8.905346078584288620943674890727, −7.34749213611770537226741174760, −6.38095633382134218879814941976, −5.26328586233941450851222247546, −4.43911128322598169191444851060, −3.70490937671561437044981485088, −3.08809539918140425934697908220, −1.80488654398499356800401042462, 1.96607414579134336926880466710, 3.02964510995801303773866080692, 3.60835081971255033958820115165, 5.26437576886206323475558330964, 5.97476543282247582651906085788, 6.51969799423836953646575037183, 7.70785488256712084057939062605, 7.961891197116207900582142011151, 9.213168689471578030513790756434, 10.18566635862932477357101138028

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