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Properties

Label 2-29e2-29.13-c1-0-8
Degree $2$
Conductor $841$
Sign $0.178 - 0.984i$
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  + (−0.983 + 2.04i)2-s + (−2.22 − 1.77i)3-s + (−1.95 − 2.45i)4-s + (0.0521 + 0.0251i)5-s + (5.80 − 2.79i)6-s + (−0.973 + 1.22i)7-s + (2.52 − 0.576i)8-s + (1.12 + 4.94i)9-s + (−0.102 + 0.0818i)10-s + (−3.87 − 0.884i)11-s + 8.92i·12-s + (0.0901 − 0.394i)13-s + (−1.53 − 3.19i)14-s + (−0.0713 − 0.148i)15-s + (0.0909 − 0.398i)16-s − 5.16i·17-s + ⋯
L(s)  = 1  + (−0.695 + 1.44i)2-s + (−1.28 − 1.02i)3-s + (−0.979 − 1.22i)4-s + (0.0233 + 0.0112i)5-s + (2.36 − 1.14i)6-s + (−0.368 + 0.461i)7-s + (0.893 − 0.203i)8-s + (0.375 + 1.64i)9-s + (−0.0324 + 0.0258i)10-s + (−1.16 − 0.266i)11-s + 2.57i·12-s + (0.0249 − 0.109i)13-s + (−0.410 − 0.852i)14-s + (−0.0184 − 0.0382i)15-s + (0.0227 − 0.0996i)16-s − 1.25i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.178 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.178 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.178 - 0.984i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.178 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.275879 + 0.230437i\)
\(L(\frac12)\) \(\approx\) \(0.275879 + 0.230437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.983 - 2.04i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (2.22 + 1.77i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (-0.0521 - 0.0251i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (0.973 - 1.22i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (3.87 + 0.884i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.0901 + 0.394i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + 5.16iT - 17T^{2} \)
19 \( 1 + (2.69 - 2.15i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (0.207 - 0.0998i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (1.87 - 3.89i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-5.23 + 1.19i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + 1.46iT - 41T^{2} \)
43 \( 1 + (-2.81 - 5.84i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-9.33 - 2.13i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-1.61 - 0.776i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 1.05T + 59T^{2} \)
61 \( 1 + (-2.49 - 1.98i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (-2.41 - 10.5i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-2.99 + 13.1i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-3.04 - 6.32i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (4.66 - 1.06i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-5.25 - 6.58i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-0.491 + 1.02i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (-13.4 + 10.7i)T + (21.5 - 94.5i)T^{2} \)
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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

−10.29274116244708535835448462732, −9.353769878391222476824802392346, −8.323061538765264198337156705247, −7.59650224288747404833009716992, −6.99683686743180387631191162432, −6.01695612589659239441534065656, −5.72988480026286765551247772355, −4.78915947628529900453391667691, −2.54726489836763314468023277781, −0.65307331289131310216084495979, 0.46447049021191595500334578066, 2.18953084379750791484051721467, 3.62573558660469801144786697267, 4.28954678795612027461383532329, 5.42443344634343582605591904815, 6.32399810548944739186889138045, 7.67390937945295745934981484654, 8.791888100841630353512450761257, 9.691257567208901051135669203130, 10.23898855653644356177235540061

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