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Properties

Label 2-29e2-29.23-c1-0-7
Degree $2$
Conductor $841$
Sign $-0.971 + 0.236i$
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  + (−1.57 + 1.96i)2-s + (0.233 − 1.02i)3-s + (−0.967 − 4.23i)4-s + (−0.913 + 1.14i)5-s + (1.65 + 2.06i)6-s + (−0.918 + 4.02i)7-s + (5.32 + 2.56i)8-s + (1.70 + 0.822i)9-s + (−0.821 − 3.59i)10-s + (−0.453 + 0.218i)11-s − 4.56·12-s + (2.81 − 1.35i)13-s + (−6.48 − 8.12i)14-s + (0.959 + 1.20i)15-s + (−5.58 + 2.68i)16-s − 2.36·17-s + ⋯
L(s)  = 1  + (−1.11 + 1.39i)2-s + (0.134 − 0.591i)3-s + (−0.483 − 2.11i)4-s + (−0.408 + 0.512i)5-s + (0.673 + 0.844i)6-s + (−0.347 + 1.52i)7-s + (1.88 + 0.906i)8-s + (0.569 + 0.274i)9-s + (−0.259 − 1.13i)10-s + (−0.136 + 0.0658i)11-s − 1.31·12-s + (0.780 − 0.375i)13-s + (−1.73 − 2.17i)14-s + (0.247 + 0.310i)15-s + (−1.39 + 0.672i)16-s − 0.573·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.971 + 0.236i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (574, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ -0.971 + 0.236i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0674625 - 0.561318i\)
\(L(\frac12)\) \(\approx\) \(0.0674625 - 0.561318i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.57 - 1.96i)T + (-0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.233 + 1.02i)T + (-2.70 - 1.30i)T^{2} \)
5 \( 1 + (0.913 - 1.14i)T + (-1.11 - 4.87i)T^{2} \)
7 \( 1 + (0.918 - 4.02i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (0.453 - 0.218i)T + (6.85 - 8.60i)T^{2} \)
13 \( 1 + (-2.81 + 1.35i)T + (8.10 - 10.1i)T^{2} \)
17 \( 1 + 2.36T + 17T^{2} \)
19 \( 1 + (0.246 + 1.07i)T + (-17.1 + 8.24i)T^{2} \)
23 \( 1 + (-2.45 - 3.07i)T + (-5.11 + 22.4i)T^{2} \)
31 \( 1 + (0.725 - 0.909i)T + (-6.89 - 30.2i)T^{2} \)
37 \( 1 + (2.76 + 1.33i)T + (23.0 + 28.9i)T^{2} \)
41 \( 1 + 9.27T + 41T^{2} \)
43 \( 1 + (-6.54 - 8.20i)T + (-9.56 + 41.9i)T^{2} \)
47 \( 1 + (0.196 - 0.0948i)T + (29.3 - 36.7i)T^{2} \)
53 \( 1 + (7.48 - 9.38i)T + (-11.7 - 51.6i)T^{2} \)
59 \( 1 + 9.66T + 59T^{2} \)
61 \( 1 + (0.202 - 0.885i)T + (-54.9 - 26.4i)T^{2} \)
67 \( 1 + (8.19 + 3.94i)T + (41.7 + 52.3i)T^{2} \)
71 \( 1 + (3.26 - 1.57i)T + (44.2 - 55.5i)T^{2} \)
73 \( 1 + (8.13 + 10.2i)T + (-16.2 + 71.1i)T^{2} \)
79 \( 1 + (-7.95 - 3.83i)T + (49.2 + 61.7i)T^{2} \)
83 \( 1 + (0.327 + 1.43i)T + (-74.7 + 36.0i)T^{2} \)
89 \( 1 + (9.59 - 12.0i)T + (-19.8 - 86.7i)T^{2} \)
97 \( 1 + (-2.06 - 9.06i)T + (-87.3 + 42.0i)T^{2} \)
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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

−10.42299899082708641629856728638, −9.316699816766089197150172838659, −8.873789551017091068222099503531, −7.973066377277751171379854916681, −7.36140248203924791527888922296, −6.52007599157158877875334061772, −5.89786642673186203364950214188, −4.88387143633700777548536369367, −3.02769901229068222560903495869, −1.56311354846897032616428421269, 0.42692115952432464113837509664, 1.55584429084842225709169607377, 3.24038822719498226678506512355, 3.98402940758633316027602874861, 4.57537943650093212911420176567, 6.63872035290086811025853412211, 7.49672435560617206761872619272, 8.517209803755550095602080900636, 9.031470137837806337789610603774, 10.01082581842902499184968955089

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