Properties

Degree 2
Conductor 29
Sign $-0.337 + 0.941i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.32 − 0.814i)2-s + (0.184 − 1.63i)3-s + (1.62 + 1.29i)4-s + (−3.83 − 7.95i)5-s + (−1.75 + 3.65i)6-s + (2.23 + 2.80i)7-s + (2.52 + 4.01i)8-s + (6.13 + 1.40i)9-s + (2.43 + 21.6i)10-s + (4.04 − 6.43i)11-s + (2.41 − 2.41i)12-s + (−7.19 + 1.64i)13-s + (−2.92 − 8.35i)14-s + (−13.7 + 4.79i)15-s + (−4.44 − 19.4i)16-s + (1.60 + 1.60i)17-s + ⋯
L(s)  = 1  + (−1.16 − 0.407i)2-s + (0.0613 − 0.544i)3-s + (0.405 + 0.323i)4-s + (−0.766 − 1.59i)5-s + (−0.293 + 0.608i)6-s + (0.319 + 0.400i)7-s + (0.315 + 0.501i)8-s + (0.681 + 0.155i)9-s + (0.243 + 2.16i)10-s + (0.367 − 0.585i)11-s + (0.201 − 0.201i)12-s + (−0.553 + 0.126i)13-s + (−0.208 − 0.596i)14-s + (−0.914 + 0.319i)15-s + (−0.278 − 1.21i)16-s + (0.0945 + 0.0945i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.337 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.337 + 0.941i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (8, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ -0.337 + 0.941i)$
$L(\frac{3}{2})$  $\approx$  $0.306532 - 0.435399i$
$L(\frac12)$  $\approx$  $0.306532 - 0.435399i$
$L(2)$   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 \( 1 + (-14.4 - 25.1i)T \)
good2 \( 1 + (2.32 + 0.814i)T + (3.12 + 2.49i)T^{2} \)
3 \( 1 + (-0.184 + 1.63i)T + (-8.77 - 2.00i)T^{2} \)
5 \( 1 + (3.83 + 7.95i)T + (-15.5 + 19.5i)T^{2} \)
7 \( 1 + (-2.23 - 2.80i)T + (-10.9 + 47.7i)T^{2} \)
11 \( 1 + (-4.04 + 6.43i)T + (-52.4 - 109. i)T^{2} \)
13 \( 1 + (7.19 - 1.64i)T + (152. - 73.3i)T^{2} \)
17 \( 1 + (-1.60 - 1.60i)T + 289iT^{2} \)
19 \( 1 + (-33.1 + 3.72i)T + (351. - 80.3i)T^{2} \)
23 \( 1 + (20.4 + 9.84i)T + (329. + 413. i)T^{2} \)
31 \( 1 + (-5.30 - 1.85i)T + (751. + 599. i)T^{2} \)
37 \( 1 + (-8.64 - 13.7i)T + (-593. + 1.23e3i)T^{2} \)
41 \( 1 + (42.1 - 42.1i)T - 1.68e3iT^{2} \)
43 \( 1 + (-1.74 - 4.98i)T + (-1.44e3 + 1.15e3i)T^{2} \)
47 \( 1 + (-60.4 - 37.9i)T + (958. + 1.99e3i)T^{2} \)
53 \( 1 + (78.2 - 37.7i)T + (1.75e3 - 2.19e3i)T^{2} \)
59 \( 1 - 67.9T + 3.48e3T^{2} \)
61 \( 1 + (-5.72 + 50.8i)T + (-3.62e3 - 828. i)T^{2} \)
67 \( 1 + (29.4 + 6.72i)T + (4.04e3 + 1.94e3i)T^{2} \)
71 \( 1 + (-17.3 + 3.95i)T + (4.54e3 - 2.18e3i)T^{2} \)
73 \( 1 + (-29.8 + 10.4i)T + (4.16e3 - 3.32e3i)T^{2} \)
79 \( 1 + (38.0 - 23.9i)T + (2.70e3 - 5.62e3i)T^{2} \)
83 \( 1 + (-3.87 + 4.85i)T + (-1.53e3 - 6.71e3i)T^{2} \)
89 \( 1 + (19.6 + 6.88i)T + (6.19e3 + 4.93e3i)T^{2} \)
97 \( 1 + (-5.86 - 52.0i)T + (-9.17e3 + 2.09e3i)T^{2} \)
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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

−16.70102681928735474513903150681, −15.88210361585928856295838808135, −13.88487595140991633459659298395, −12.40638337809450761642924672600, −11.58914695255535930603347282109, −9.716777864205082317641877001184, −8.578482969546489390367825409902, −7.68277844623350846763864595444, −4.90580951099626488708197463179, −1.17954302663865593114647519843, 3.90072125044681505584547539059, 7.03094311528990101048999266976, 7.69922458418077626316155297013, 9.713429505391020157712454750265, 10.39504041112957960905669352727, 11.84190136319275534064095879902, 14.05862713610426704586561243096, 15.28443938567549266227053008460, 15.98616423875418319101070520477, 17.50589154895856894674935056103

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