Properties

Label 2-29e2-29.4-c1-0-20
Degree $2$
Conductor $841$
Sign $0.864 + 0.501i$
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  + (−0.403 − 0.0921i)2-s + (0.179 + 0.373i)3-s + (−1.64 − 0.793i)4-s + (−0.222 + 0.974i)5-s + (−0.0381 − 0.167i)6-s + (−2.54 + 1.22i)7-s + (1.23 + 0.988i)8-s + (1.76 − 2.21i)9-s + (0.179 − 0.373i)10-s + (−1.88 + 1.50i)11-s − 0.757i·12-s + (−1.14 − 1.42i)13-s + (1.14 − 0.260i)14-s + (−0.403 + 0.0921i)15-s + (1.87 + 2.34i)16-s − 4.82i·17-s + ⋯
L(s)  = 1  + (−0.285 − 0.0651i)2-s + (0.103 + 0.215i)3-s + (−0.823 − 0.396i)4-s + (−0.0995 + 0.436i)5-s + (−0.0155 − 0.0682i)6-s + (−0.963 + 0.463i)7-s + (0.438 + 0.349i)8-s + (0.587 − 0.737i)9-s + (0.0568 − 0.118i)10-s + (−0.569 + 0.453i)11-s − 0.218i·12-s + (−0.316 − 0.396i)13-s + (0.305 − 0.0696i)14-s + (−0.104 + 0.0237i)15-s + (0.467 + 0.586i)16-s − 1.17i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.864 + 0.501i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (236, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.864 + 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886012 - 0.238381i\)
\(L(\frac12)\) \(\approx\) \(0.886012 - 0.238381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.403 + 0.0921i)T + (1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.179 - 0.373i)T + (-1.87 + 2.34i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (2.54 - 1.22i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (1.88 - 1.50i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.14 + 1.42i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 4.82iT - 17T^{2} \)
19 \( 1 + (-2.60 + 5.40i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (-1.70 - 7.46i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (-3.96 - 0.905i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-3.12 - 2.49i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 + 12.4iT - 41T^{2} \)
43 \( 1 + (-6.25 + 1.42i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-4.09 + 3.26i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-1.66 + 7.29i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 7.65T + 59T^{2} \)
61 \( 1 + (0.359 + 0.746i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (-3.52 + 4.42i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (-1.97 - 2.47i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (3.89 - 0.890i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (-0.323 - 0.258i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (-3.29 - 1.58i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (4.37 + 0.998i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (-5.41 + 11.2i)T + (-60.4 - 75.8i)T^{2} \)
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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

−9.815087073079232290951634421312, −9.475009163714117461960521698785, −8.806360727280539183340563337256, −7.42236782579285652355635141183, −6.88897710674304654764568061957, −5.55500842436060302956577960263, −4.87196274165517336952157359836, −3.60351571433504378182332998256, −2.66517247728247423031600804322, −0.67990264209233075088477754893, 0.990344354220281144391084682524, 2.79619311446404811701296187992, 4.04563885874231151402958975269, 4.68786266736573322228473130312, 5.96360304535133451394357997973, 7.00082231843212037134429020548, 7.989342583006545635903863301703, 8.375264045923506948329687535479, 9.460969855901652924078377293947, 10.17120615522371615288529685197

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