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Properties

Label 2-29e2-29.5-c1-0-5
Degree $2$
Conductor $841$
Sign $-0.565 - 0.824i$
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  + (0.483 − 0.385i)2-s + (0.602 − 0.137i)3-s + (−0.360 + 1.57i)4-s + (−2.40 − 3.01i)5-s + (0.238 − 0.298i)6-s + (0.497 + 2.18i)7-s + (0.970 + 2.01i)8-s + (−2.35 + 1.13i)9-s + (−2.32 − 0.530i)10-s + (0.599 − 1.24i)11-s + i·12-s + (−0.212 − 0.102i)13-s + (1.08 + 0.861i)14-s + (−1.86 − 1.48i)15-s + (−1.67 − 0.804i)16-s + 4.38i·17-s + ⋯
L(s)  = 1  + (0.341 − 0.272i)2-s + (0.347 − 0.0794i)3-s + (−0.180 + 0.788i)4-s + (−1.07 − 1.34i)5-s + (0.0972 − 0.121i)6-s + (0.188 + 0.823i)7-s + (0.343 + 0.712i)8-s + (−0.786 + 0.378i)9-s + (−0.734 − 0.167i)10-s + (0.180 − 0.375i)11-s + 0.288i·12-s + (−0.0589 − 0.0284i)13-s + (0.288 + 0.230i)14-s + (−0.480 − 0.383i)15-s + (−0.417 − 0.201i)16-s + 1.06i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.565 - 0.824i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(841\)    =    \(29^{2}\)
Sign: $-0.565 - 0.824i$
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: \((2,\ 841,\ (\ :1/2),\ -0.565 - 0.824i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.306850 + 0.582156i\)
\(L(\frac12)\) \(\approx\) \(0.306850 + 0.582156i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.483 + 0.385i)T + (0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.602 + 0.137i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (2.40 + 3.01i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (-0.497 - 2.18i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (-0.599 + 1.24i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (0.212 + 0.102i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 4.38iT - 17T^{2} \)
19 \( 1 + (4.73 + 1.08i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (0.770 - 0.966i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (7.88 - 6.29i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (2.04 + 4.24i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 3.85iT - 41T^{2} \)
43 \( 1 + (-5.65 - 4.51i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (3.03 - 6.30i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (1.24 + 1.56i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 6.09T + 59T^{2} \)
61 \( 1 + (-0.602 + 0.137i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (1.37 - 0.662i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (-9.43 - 4.54i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (10.7 + 8.54i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (2.64 + 5.48i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (-2.21 + 9.69i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (-3.68 + 2.93i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (3.47 + 0.792i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.90510885836201167127089234412, −9.111267277900372621937855259255, −8.645078161733281963902768104852, −8.252756480572769723726063788019, −7.45783737909508928540323009919, −5.84652687668380967870102493105, −4.95663990893561782618482424371, −4.10515849839985216788114537371, −3.23249779465590422312468948117, −1.93300539523557258505901627711, 0.26276323696761564905778988739, 2.38899737439150395543751648070, 3.70529036693392654659806046813, 4.21622049898435331068550929819, 5.54210754087037635665342341305, 6.65590921620961194637362262718, 7.13200685429224085106084946167, 8.007961975748728325412589834558, 9.118905893325525693517710241744, 10.05313508521739053352009388103

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