Properties

Degree 2
Conductor 29
Sign $0.638 - 0.769i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.29 + 0.811i)2-s + (2.15 + 0.752i)3-s + (−0.725 + 1.50i)4-s + (3.36 + 0.767i)5-s + (−3.38 + 0.773i)6-s + (−0.255 + 0.123i)7-s + (−0.968 − 8.59i)8-s + (−2.97 − 2.37i)9-s + (−4.96 + 1.73i)10-s + (1.62 − 14.4i)11-s + (−2.69 + 2.69i)12-s + (−4.30 + 3.43i)13-s + (0.230 − 0.366i)14-s + (6.65 + 4.18i)15-s + (4.06 + 5.09i)16-s + (5.25 + 5.25i)17-s + ⋯
L(s)  = 1  + (−0.645 + 0.405i)2-s + (0.716 + 0.250i)3-s + (−0.181 + 0.376i)4-s + (0.672 + 0.153i)5-s + (−0.564 + 0.128i)6-s + (−0.0365 + 0.0175i)7-s + (−0.121 − 1.07i)8-s + (−0.330 − 0.263i)9-s + (−0.496 + 0.173i)10-s + (0.147 − 1.31i)11-s + (−0.224 + 0.224i)12-s + (−0.331 + 0.264i)13-s + (0.0164 − 0.0261i)14-s + (0.443 + 0.278i)15-s + (0.253 + 0.318i)16-s + (0.309 + 0.309i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.638 - 0.769i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.638 - 0.769i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.638 - 0.769i)$
$L(\frac{3}{2})$  $\approx$  $0.785214 + 0.368712i$
$L(\frac12)$  $\approx$  $0.785214 + 0.368712i$
$L(2)$   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 \( 1 + (-2.32 - 28.9i)T \)
good2 \( 1 + (1.29 - 0.811i)T + (1.73 - 3.60i)T^{2} \)
3 \( 1 + (-2.15 - 0.752i)T + (7.03 + 5.61i)T^{2} \)
5 \( 1 + (-3.36 - 0.767i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (0.255 - 0.123i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (-1.62 + 14.4i)T + (-117. - 26.9i)T^{2} \)
13 \( 1 + (4.30 - 3.43i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (-5.25 - 5.25i)T + 289iT^{2} \)
19 \( 1 + (-9.56 - 27.3i)T + (-282. + 225. i)T^{2} \)
23 \( 1 + (6.05 + 26.5i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (12.9 - 8.13i)T + (416. - 865. i)T^{2} \)
37 \( 1 + (-3.05 - 27.1i)T + (-1.33e3 + 304. i)T^{2} \)
41 \( 1 + (45.8 - 45.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (-35.9 + 57.2i)T + (-802. - 1.66e3i)T^{2} \)
47 \( 1 + (-17.2 - 1.94i)T + (2.15e3 + 491. i)T^{2} \)
53 \( 1 + (-1.31 + 5.74i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 + 43.1T + 3.48e3T^{2} \)
61 \( 1 + (-104. - 36.6i)T + (2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (67.4 + 53.7i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (-45.1 + 36.0i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (42.0 + 26.4i)T + (2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (-96.9 + 10.9i)T + (6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (0.956 + 0.460i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (-73.6 + 46.2i)T + (3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (-54.6 + 19.1i)T + (7.35e3 - 5.86e3i)T^{2} \)
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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

−16.97982848506661184394450180649, −16.19154670989613146986927138345, −14.56413430881235410062541524901, −13.71765019770498667703813087266, −12.16054559921068263722670289020, −10.18256031598930259765344455202, −9.010328069823498036991138808595, −8.076113255426539016348120229707, −6.19220692096271339542401984867, −3.43132571465032652387404096427, 2.17461356851085325180502795350, 5.30783691137417245000378221649, 7.57894634227678608785125029456, 9.175641038557165114019145967336, 9.881759917484207708094080712341, 11.48973416065095789675545724457, 13.24997571189410904408118477451, 14.19594789449660392059624167243, 15.32328571713014875358297870437, 17.32069491645425100754151473007

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