Properties

Degree 2
Conductor 29
Sign $-0.571 - 0.820i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63 + 2.59i)2-s + (0.532 + 1.52i)3-s + (−2.35 − 4.88i)4-s + (−4.43 + 1.01i)5-s + (−4.81 − 1.10i)6-s + (10.7 + 5.18i)7-s + (4.32 + 0.487i)8-s + (5.00 − 3.99i)9-s + (4.61 − 13.1i)10-s + (2.90 − 0.326i)11-s + (6.17 − 6.17i)12-s + (2.24 + 1.79i)13-s + (−31.0 + 19.5i)14-s + (−3.90 − 6.21i)15-s + (5.18 − 6.49i)16-s + (2.44 + 2.44i)17-s + ⋯
L(s)  = 1  + (−0.816 + 1.29i)2-s + (0.177 + 0.506i)3-s + (−0.587 − 1.22i)4-s + (−0.887 + 0.202i)5-s + (−0.803 − 0.183i)6-s + (1.53 + 0.740i)7-s + (0.540 + 0.0609i)8-s + (0.556 − 0.443i)9-s + (0.461 − 1.31i)10-s + (0.263 − 0.0297i)11-s + (0.514 − 0.514i)12-s + (0.173 + 0.137i)13-s + (−2.21 + 1.39i)14-s + (−0.260 − 0.414i)15-s + (0.323 − 0.406i)16-s + (0.144 + 0.144i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.571 - 0.820i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (26, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ -0.571 - 0.820i)$
$L(\frac{3}{2})$  $\approx$  $0.320825 + 0.614390i$
$L(\frac12)$  $\approx$  $0.320825 + 0.614390i$
$L(2)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 29$, \(F_p\) is a polynomial of degree 2. If $p = 29$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad29 \( 1 + (-18.6 - 22.1i)T \)
good2 \( 1 + (1.63 - 2.59i)T + (-1.73 - 3.60i)T^{2} \)
3 \( 1 + (-0.532 - 1.52i)T + (-7.03 + 5.61i)T^{2} \)
5 \( 1 + (4.43 - 1.01i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + (-10.7 - 5.18i)T + (30.5 + 38.3i)T^{2} \)
11 \( 1 + (-2.90 + 0.326i)T + (117. - 26.9i)T^{2} \)
13 \( 1 + (-2.24 - 1.79i)T + (37.6 + 164. i)T^{2} \)
17 \( 1 + (-2.44 - 2.44i)T + 289iT^{2} \)
19 \( 1 + (31.9 + 11.1i)T + (282. + 225. i)T^{2} \)
23 \( 1 + (-9.24 + 40.5i)T + (-476. - 229. i)T^{2} \)
31 \( 1 + (-7.40 + 11.7i)T + (-416. - 865. i)T^{2} \)
37 \( 1 + (19.6 + 2.21i)T + (1.33e3 + 304. i)T^{2} \)
41 \( 1 + (15.2 - 15.2i)T - 1.68e3iT^{2} \)
43 \( 1 + (-1.58 + 0.996i)T + (802. - 1.66e3i)T^{2} \)
47 \( 1 + (0.858 + 7.61i)T + (-2.15e3 + 491. i)T^{2} \)
53 \( 1 + (7.13 + 31.2i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 + 9.47T + 3.48e3T^{2} \)
61 \( 1 + (19.5 + 55.9i)T + (-2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (45.3 - 36.1i)T + (998. - 4.37e3i)T^{2} \)
71 \( 1 + (61.2 + 48.8i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (-40.7 - 64.8i)T + (-2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (-10.0 + 89.0i)T + (-6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (-42.5 + 20.4i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (31.5 - 50.2i)T + (-3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (41.8 - 119. i)T + (-7.35e3 - 5.86e3i)T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−17.23856256126877433190197218558, −15.99203046753168301764906052713, −14.96467020025504869563746499045, −14.72953442183703781293907322263, −12.17515525933938461603735878661, −10.71877835002087893653473605353, −8.889760640563614695385713131598, −8.181868709471769028592558954771, −6.67042359836006166507770548578, −4.62841176871446454317879243770, 1.54582187350455528357458628921, 4.20000897227536958987557539224, 7.63609875391988659152222184250, 8.450760777072034907520870050397, 10.35120613074008766428032257735, 11.31884799898038880041574579505, 12.28616304868042526041411175560, 13.70248414465473620631074328250, 15.26117592970920691074343045702, 17.07158005106350380119730704774

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