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Properties

Label 2-29e2-29.6-c1-0-23
Degree $2$
Conductor $841$
Sign $-0.565 + 0.824i$
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  + (−1.26 − 1.00i)2-s + (−1.57 − 0.360i)3-s + (0.137 + 0.602i)4-s + (1.77 − 2.23i)5-s + (1.63 + 2.04i)6-s + (−0.497 + 2.18i)7-s + (−0.970 + 2.01i)8-s + (−0.344 − 0.165i)9-s + (−4.50 + 1.02i)10-s + (1.56 + 3.25i)11-s − 1.00i·12-s + (3.81 − 1.83i)13-s + (2.82 − 2.25i)14-s + (−3.61 + 2.87i)15-s + (4.37 − 2.10i)16-s − 6.61i·17-s + ⋯
L(s)  = 1  + (−0.894 − 0.713i)2-s + (−0.910 − 0.207i)3-s + (0.0687 + 0.301i)4-s + (0.795 − 0.997i)5-s + (0.666 + 0.835i)6-s + (−0.188 + 0.823i)7-s + (−0.343 + 0.712i)8-s + (−0.114 − 0.0552i)9-s + (−1.42 + 0.324i)10-s + (0.473 + 0.982i)11-s − 0.288i·12-s + (1.05 − 0.509i)13-s + (0.755 − 0.602i)14-s + (−0.932 + 0.743i)15-s + (1.09 − 0.526i)16-s − 1.60i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 + 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.565 + 0.824i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.565 + 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.342283 - 0.649381i\)
\(L(\frac12)\) \(\approx\) \(0.342283 - 0.649381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.26 + 1.00i)T + (0.445 + 1.94i)T^{2} \)
3 \( 1 + (1.57 + 0.360i)T + (2.70 + 1.30i)T^{2} \)
5 \( 1 + (-1.77 + 2.23i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.497 - 2.18i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-1.56 - 3.25i)T + (-6.85 + 8.60i)T^{2} \)
13 \( 1 + (-3.81 + 1.83i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 6.61iT - 17T^{2} \)
19 \( 1 + (-1.80 + 0.412i)T + (17.1 - 8.24i)T^{2} \)
23 \( 1 + (-2.01 - 2.53i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (-0.852 - 0.679i)T + (6.89 + 30.2i)T^{2} \)
37 \( 1 + (-3.77 + 7.84i)T + (-23.0 - 28.9i)T^{2} \)
41 \( 1 - 2.85iT - 41T^{2} \)
43 \( 1 + (-2.16 + 1.72i)T + (9.56 - 41.9i)T^{2} \)
47 \( 1 + (3.03 + 6.30i)T + (-29.3 + 36.7i)T^{2} \)
53 \( 1 + (1.24 - 1.56i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 5.09T + 59T^{2} \)
61 \( 1 + (1.57 + 0.360i)T + (54.9 + 26.4i)T^{2} \)
67 \( 1 + (9.43 + 4.54i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (-1.37 + 0.662i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (0.228 - 0.181i)T + (16.2 - 71.1i)T^{2} \)
79 \( 1 + (-2.20 + 4.58i)T + (-49.2 - 61.7i)T^{2} \)
83 \( 1 + (1.76 + 7.74i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (6.80 + 5.42i)T + (19.8 + 86.7i)T^{2} \)
97 \( 1 + (-16.1 + 3.68i)T + (87.3 - 42.0i)T^{2} \)
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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

−9.733870272012415570263868411257, −9.162370842504550555677295001881, −8.788685097889606404601797141284, −7.45087056790216727870595599880, −6.15880457349184427505200127908, −5.52465015843562092721002594166, −4.88789423283685519466120277982, −2.94544649406752009660325219996, −1.68945195948312917538974593688, −0.69014234808942724009505760267, 1.12224591260586076516398375538, 3.15002091747190172084854260823, 4.14631547233233011121354194394, 5.85907380631536199808338416122, 6.29738225671778748800591900477, 6.77077286010727967194812812327, 8.007244712473191471519482848725, 8.716194459782274668449473104541, 9.705958650576483894190154543394, 10.59031447001042043720734937713

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