Properties

Degree 2
Conductor 29
Sign $0.717 - 0.697i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.42 + 2.27i)2-s + (0.724 − 2.07i)3-s + (−1.39 + 2.88i)4-s + (−6.69 − 1.52i)5-s + (5.74 − 1.31i)6-s + (−4.78 + 2.30i)7-s + (2.11 − 0.238i)8-s + (3.27 + 2.61i)9-s + (−6.09 − 17.4i)10-s + (−3.10 − 0.349i)11-s + (4.97 + 4.97i)12-s + (14.9 − 11.9i)13-s + (−12.0 − 7.57i)14-s + (−8.02 + 12.7i)15-s + (11.5 + 14.5i)16-s + (−19.1 + 19.1i)17-s + ⋯
L(s)  = 1  + (0.714 + 1.13i)2-s + (0.241 − 0.690i)3-s + (−0.347 + 0.722i)4-s + (−1.33 − 0.305i)5-s + (0.956 − 0.218i)6-s + (−0.682 + 0.328i)7-s + (0.264 − 0.0298i)8-s + (0.363 + 0.290i)9-s + (−0.609 − 1.74i)10-s + (−0.281 − 0.0317i)11-s + (0.414 + 0.414i)12-s + (1.14 − 0.915i)13-s + (−0.861 − 0.541i)14-s + (−0.534 + 0.851i)15-s + (0.722 + 0.906i)16-s + (−1.12 + 1.12i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.717 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.717 - 0.697i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (19, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ 0.717 - 0.697i)$
$L(\frac{3}{2})$  $\approx$  $1.13928 + 0.462483i$
$L(\frac12)$  $\approx$  $1.13928 + 0.462483i$
$L(2)$   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 \( 1 + (22.5 - 18.2i)T \)
good2 \( 1 + (-1.42 - 2.27i)T + (-1.73 + 3.60i)T^{2} \)
3 \( 1 + (-0.724 + 2.07i)T + (-7.03 - 5.61i)T^{2} \)
5 \( 1 + (6.69 + 1.52i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (4.78 - 2.30i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (3.10 + 0.349i)T + (117. + 26.9i)T^{2} \)
13 \( 1 + (-14.9 + 11.9i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (19.1 - 19.1i)T - 289iT^{2} \)
19 \( 1 + (-3.26 + 1.14i)T + (282. - 225. i)T^{2} \)
23 \( 1 + (3.11 + 13.6i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (30.9 + 49.2i)T + (-416. + 865. i)T^{2} \)
37 \( 1 + (-23.4 + 2.64i)T + (1.33e3 - 304. i)T^{2} \)
41 \( 1 + (-29.0 - 29.0i)T + 1.68e3iT^{2} \)
43 \( 1 + (-28.3 - 17.8i)T + (802. + 1.66e3i)T^{2} \)
47 \( 1 + (-5.42 + 48.1i)T + (-2.15e3 - 491. i)T^{2} \)
53 \( 1 + (-2.93 + 12.8i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 - 11.7T + 3.48e3T^{2} \)
61 \( 1 + (18.2 - 52.2i)T + (-2.90e3 - 2.32e3i)T^{2} \)
67 \( 1 + (6.83 + 5.44i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (49.1 - 39.1i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (-33.0 + 52.6i)T + (-2.31e3 - 4.80e3i)T^{2} \)
79 \( 1 + (-5.05 - 44.8i)T + (-6.08e3 + 1.38e3i)T^{2} \)
83 \( 1 + (123. + 59.5i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (-38.7 - 61.6i)T + (-3.43e3 + 7.13e3i)T^{2} \)
97 \( 1 + (-16.9 - 48.4i)T + (-7.35e3 + 5.86e3i)T^{2} \)
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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

−16.44928336807855201941437637174, −15.72897126988792903741666482545, −14.95156764510463607958060818374, −13.14676265693164104621548812828, −12.84628278555042872567093593408, −10.91536363681713318376553660960, −8.390620663453847311804182573687, −7.48557432808695614250748570858, −6.07221163745282262389688776946, −4.08670171568366734569995492993, 3.44172461303241243960910841954, 4.29959478948468261868990050981, 7.20386649415626310636730727760, 9.271165678538823703999266964144, 10.79575669638434183916865149891, 11.58604407007979843881385770772, 12.88632198872024851596473646810, 14.09051550036481991710050497931, 15.67413588185734315712784137574, 16.16309691269335600609777226004

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