Properties

Label 2-29e2-29.6-c1-0-15
Degree $2$
Conductor $841$
Sign $-0.160 + 0.987i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.160 + 0.987i$
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: \((2,\ 841,\ (\ :1/2),\ -0.160 + 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.304324 - 0.357858i\)
\(L(\frac12)\) \(\approx\) \(0.304324 - 0.357858i\)
\(L(\frac{3}{2})\) 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

−10.02122607813636707723587569164, −9.524751764427014563012520666359, −8.452165645061926324956262966530, −7.32606933821160318898687347642, −6.95862251161276008711624717861, −5.48160017266301799531313502776, −4.61089321462873755532434496267, −3.13252215477956549131480877625, −1.54743694664400488968375974637, −0.75618198234381513461313468474, 0.74618650975108476512765213925, 2.58714473508557471157493514455, 4.79933653257148156423393557791, 5.55778780629583880727950728327, 6.12264874605430507137313092820, 6.93990233135114037157980655149, 7.86889065824014341526188592033, 8.711297121112129120084554214187, 9.742441517290921978675908188082, 10.09760343392094699979110760799

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