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Properties

Label 2-29e2-29.5-c1-0-48
Degree $2$
Conductor $841$
Sign $0.120 + 0.992i$
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  + (1.88 − 1.50i)2-s + (2.35 − 0.537i)3-s + (0.851 − 3.73i)4-s + (0.623 + 0.781i)5-s + (3.63 − 4.55i)6-s + (0.629 + 2.75i)7-s + (−1.91 − 3.97i)8-s + (2.54 − 1.22i)9-s + (2.35 + 0.537i)10-s + (−0.179 + 0.373i)11-s − 9.24i·12-s + (−3.44 − 1.66i)13-s + (5.33 + 4.25i)14-s + (1.88 + 1.50i)15-s + (−2.70 − 1.30i)16-s + 0.828i·17-s + ⋯
L(s)  = 1  + (1.33 − 1.06i)2-s + (1.35 − 0.310i)3-s + (0.425 − 1.86i)4-s + (0.278 + 0.349i)5-s + (1.48 − 1.86i)6-s + (0.237 + 1.04i)7-s + (−0.677 − 1.40i)8-s + (0.849 − 0.409i)9-s + (0.744 + 0.169i)10-s + (−0.0541 + 0.112i)11-s − 2.66i·12-s + (−0.956 − 0.460i)13-s + (1.42 + 1.13i)14-s + (0.487 + 0.388i)15-s + (−0.675 − 0.325i)16-s + 0.200i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.120 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $0.120 + 0.992i$
Analytic conductor: \(6.71541\)
Root analytic conductor: \(2.59141\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{841} (63, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 841,\ (\ :1/2),\ 0.120 + 0.992i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.61741 - 3.20576i\)
\(L(\frac12)\) \(\approx\) \(3.61741 - 3.20576i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.88 + 1.50i)T + (0.445 - 1.94i)T^{2} \)
3 \( 1 + (-2.35 + 0.537i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (-0.623 - 0.781i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.629 - 2.75i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (0.179 - 0.373i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (3.44 + 1.66i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 0.828iT - 17T^{2} \)
19 \( 1 + (5.84 + 1.33i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-2.28 + 2.85i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (7.87 - 6.27i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (1.73 + 3.60i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 4.48iT - 41T^{2} \)
43 \( 1 + (-2.80 - 2.23i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-1.40 + 2.92i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (-5.91 - 7.41i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 + 3.65T + 59T^{2} \)
61 \( 1 + (4.70 - 1.07i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (-5.09 + 2.45i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (7.95 + 3.83i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (3.12 + 2.49i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (-1.04 - 2.17i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (1.70 - 7.46i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-9.76 + 7.78i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (-4.37 - 0.998i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.28704969109148338200249163553, −9.148734981480954139235565909001, −8.553461619429154219068455630045, −7.42414502166723461733053853060, −6.27996222083041315324102857702, −5.29595134064849361606805988941, −4.37074839047684555623437926538, −3.17611429133733178584901011323, −2.49051656211348586196167346348, −1.95149536725459685652071588749, 2.14269768273526532893404942033, 3.43571702097507796280295127588, 4.11918533764253722950617541115, 4.87847712029804537895414477540, 5.92280277588152029124310725184, 7.21523278002160156568242134181, 7.46868522337646355860608655616, 8.508866761820175972085502010084, 9.284347553272043987608532159650, 10.22043641843770688698015646701

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