Properties

Degree 2
Conductor 29
Sign $0.465 + 0.885i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.488 − 0.776i)2-s + (−1.66 − 4.74i)3-s + (1.37 + 2.84i)4-s + (−1.98 + 0.453i)5-s + (−4.50 − 1.02i)6-s + (9.56 + 4.60i)7-s + (6.52 + 0.735i)8-s + (−12.7 + 10.1i)9-s + (−0.616 + 1.76i)10-s + (−11.6 + 1.31i)11-s + (11.2 − 11.2i)12-s + (−11.0 − 8.79i)13-s + (8.24 − 5.18i)14-s + (5.45 + 8.67i)15-s + (−4.12 + 5.16i)16-s + (0.154 + 0.154i)17-s + ⋯
L(s)  = 1  + (0.244 − 0.388i)2-s + (−0.553 − 1.58i)3-s + (0.342 + 0.711i)4-s + (−0.396 + 0.0906i)5-s + (−0.750 − 0.171i)6-s + (1.36 + 0.657i)7-s + (0.815 + 0.0919i)8-s + (−1.41 + 1.13i)9-s + (−0.0616 + 0.176i)10-s + (−1.05 + 0.119i)11-s + (0.936 − 0.936i)12-s + (−0.848 − 0.676i)13-s + (0.588 − 0.370i)14-s + (0.363 + 0.578i)15-s + (−0.257 + 0.323i)16-s + (0.00910 + 0.00910i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.465 + 0.885i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.465 + 0.885i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (26, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.465 + 0.885i)$
$L(\frac{3}{2})$  $\approx$  $0.859544 - 0.519135i$
$L(\frac12)$  $\approx$  $0.859544 - 0.519135i$
$L(2)$   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 \( 1 + (15.7 + 24.3i)T \)
good2 \( 1 + (-0.488 + 0.776i)T + (-1.73 - 3.60i)T^{2} \)
3 \( 1 + (1.66 + 4.74i)T + (-7.03 + 5.61i)T^{2} \)
5 \( 1 + (1.98 - 0.453i)T + (22.5 - 10.8i)T^{2} \)
7 \( 1 + (-9.56 - 4.60i)T + (30.5 + 38.3i)T^{2} \)
11 \( 1 + (11.6 - 1.31i)T + (117. - 26.9i)T^{2} \)
13 \( 1 + (11.0 + 8.79i)T + (37.6 + 164. i)T^{2} \)
17 \( 1 + (-0.154 - 0.154i)T + 289iT^{2} \)
19 \( 1 + (-20.3 - 7.12i)T + (282. + 225. i)T^{2} \)
23 \( 1 + (-1.51 + 6.62i)T + (-476. - 229. i)T^{2} \)
31 \( 1 + (0.406 - 0.646i)T + (-416. - 865. i)T^{2} \)
37 \( 1 + (6.96 + 0.784i)T + (1.33e3 + 304. i)T^{2} \)
41 \( 1 + (-35.8 + 35.8i)T - 1.68e3iT^{2} \)
43 \( 1 + (-18.1 + 11.4i)T + (802. - 1.66e3i)T^{2} \)
47 \( 1 + (2.57 + 22.8i)T + (-2.15e3 + 491. i)T^{2} \)
53 \( 1 + (-12.7 - 55.8i)T + (-2.53e3 + 1.21e3i)T^{2} \)
59 \( 1 + 48.4T + 3.48e3T^{2} \)
61 \( 1 + (24.4 + 69.8i)T + (-2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (-33.7 + 26.9i)T + (998. - 4.37e3i)T^{2} \)
71 \( 1 + (-78.3 - 62.5i)T + (1.12e3 + 4.91e3i)T^{2} \)
73 \( 1 + (-5.59 - 8.90i)T + (-2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (-4.65 + 41.2i)T + (-6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (-85.5 + 41.1i)T + (4.29e3 - 5.38e3i)T^{2} \)
89 \( 1 + (89.2 - 142. i)T + (-3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (-20.2 + 57.9i)T + (-7.35e3 - 5.86e3i)T^{2} \)
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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

−17.16850074450450267801836593695, −15.47498200219575080012435292779, −13.83770272992953111112053281161, −12.59170739203885546360805801045, −11.93698043641229834294827352861, −11.01946568590702735627916149109, −7.914172458501602068251131731395, −7.56610913536133841216382978870, −5.37205647845159185847971192468, −2.26853040547057054155404056923, 4.52504164191194408946065898757, 5.33475322398810176268929386281, 7.58632643514464683914865254181, 9.686240590628959892687459855479, 10.81102061500491318263156519581, 11.50581895660908274251584403118, 14.03669515958099033294414169551, 14.93922157394777123780629646483, 15.86031556613409573045068230384, 16.67361498564826132023047509789

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