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Properties

Label 2-29e2-29.9-c1-0-41
Degree $2$
Conductor $841$
Sign $-0.860 + 0.509i$
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  + (−0.193 − 0.400i)2-s + (−0.974 + 0.777i)3-s + (1.12 − 1.40i)4-s + (0.623 − 0.300i)5-s + (0.5 + 0.240i)6-s + (0.222 + 0.279i)7-s + (−1.64 − 0.376i)8-s + (−0.321 + 1.40i)9-s + (−0.240 − 0.192i)10-s + (−4.81 + 1.09i)11-s + 2.24i·12-s + (−1.25 − 5.51i)13-s + (0.0689 − 0.143i)14-s + (−0.374 + 0.777i)15-s + (−0.634 − 2.77i)16-s − 4.49i·17-s + ⋯
L(s)  = 1  + (−0.136 − 0.283i)2-s + (−0.562 + 0.448i)3-s + (0.561 − 0.704i)4-s + (0.278 − 0.134i)5-s + (0.204 + 0.0983i)6-s + (0.0841 + 0.105i)7-s + (−0.583 − 0.133i)8-s + (−0.107 + 0.469i)9-s + (−0.0761 − 0.0607i)10-s + (−1.45 + 0.331i)11-s + 0.648i·12-s + (−0.348 − 1.52i)13-s + (0.0184 − 0.0382i)14-s + (−0.0966 + 0.200i)15-s + (−0.158 − 0.694i)16-s − 1.08i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.860 + 0.509i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (270, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.167201 - 0.610205i\)
\(L(\frac12)\) \(\approx\) \(0.167201 - 0.610205i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.193 + 0.400i)T + (-1.24 + 1.56i)T^{2} \)
3 \( 1 + (0.974 - 0.777i)T + (0.667 - 2.92i)T^{2} \)
5 \( 1 + (-0.623 + 0.300i)T + (3.11 - 3.90i)T^{2} \)
7 \( 1 + (-0.222 - 0.279i)T + (-1.55 + 6.82i)T^{2} \)
11 \( 1 + (4.81 - 1.09i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.25 + 5.51i)T + (-11.7 + 5.64i)T^{2} \)
17 \( 1 + 4.49iT - 17T^{2} \)
19 \( 1 + (-1.84 - 1.46i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (2.06 + 0.996i)T + (14.3 + 17.9i)T^{2} \)
31 \( 1 + (2.90 + 6.02i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (-4.81 - 1.09i)T + (33.3 + 16.0i)T^{2} \)
41 \( 1 - 3.10iT - 41T^{2} \)
43 \( 1 + (-1.47 + 3.06i)T + (-26.8 - 33.6i)T^{2} \)
47 \( 1 + (6.28 - 1.43i)T + (42.3 - 20.3i)T^{2} \)
53 \( 1 + (-4.22 + 2.03i)T + (33.0 - 41.4i)T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + (1.28 - 1.02i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (0.516 - 2.26i)T + (-60.3 - 29.0i)T^{2} \)
71 \( 1 + (1.63 + 7.15i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (-2.44 + 5.06i)T + (-45.5 - 57.0i)T^{2} \)
79 \( 1 + (4.54 + 1.03i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (-2.77 + 3.48i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (2.46 + 5.11i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (0.141 + 0.112i)T + (21.5 + 94.5i)T^{2} \)
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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

−10.06145370619828945359673824888, −9.457479295096845776474055015759, −7.941829029443197609870349692411, −7.46887898660728959004476984942, −5.96505820494676622432320559384, −5.41972183857653003870174965745, −4.85250168338350593602225879681, −3.01245053165992360767291516672, −2.13477209152695125186095799025, −0.31201918126235598848028139388, 1.85391432958332946431327488479, 3.00291655607918465600594840462, 4.23353432410333379559359219459, 5.61839133112306491369585817634, 6.31622031410905661784203635747, 7.07732265510674132850828186791, 7.83846889139999141297364228191, 8.705852441425647885028727062274, 9.663619772945368127765337205790, 10.77828730539887426660335017185

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