Properties

Degree 2
Conductor 29
Sign $-0.681 + 0.731i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.24 + 2.03i)2-s + (−2.23 − 0.780i)3-s + (4.63 − 9.61i)4-s + (−5.12 − 1.17i)5-s + (8.82 − 2.01i)6-s + (−6.56 + 3.16i)7-s + (2.86 + 25.4i)8-s + (−2.67 − 2.13i)9-s + (19.0 − 6.65i)10-s + (−0.480 + 4.26i)11-s + (−17.8 + 17.8i)12-s + (2.73 − 2.17i)13-s + (14.8 − 23.6i)14-s + (10.5 + 6.61i)15-s + (−34.4 − 43.2i)16-s + (−6.15 − 6.15i)17-s + ⋯
L(s)  = 1  + (−1.62 + 1.01i)2-s + (−0.743 − 0.260i)3-s + (1.15 − 2.40i)4-s + (−1.02 − 0.234i)5-s + (1.47 − 0.335i)6-s + (−0.938 + 0.451i)7-s + (0.357 + 3.17i)8-s + (−0.296 − 0.236i)9-s + (1.90 − 0.665i)10-s + (−0.0437 + 0.387i)11-s + (−1.48 + 1.48i)12-s + (0.210 − 0.167i)13-s + (1.06 − 1.68i)14-s + (0.701 + 0.440i)15-s + (−2.15 − 2.70i)16-s + (−0.362 − 0.362i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.681 + 0.731i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.681 + 0.731i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (10, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ -0.681 + 0.731i)$
$L(\frac{3}{2})$  $\approx$  $0.00916210 - 0.0210649i$
$L(\frac12)$  $\approx$  $0.00916210 - 0.0210649i$
$L(2)$   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 \( 1 + (0.661 + 28.9i)T \)
good2 \( 1 + (3.24 - 2.03i)T + (1.73 - 3.60i)T^{2} \)
3 \( 1 + (2.23 + 0.780i)T + (7.03 + 5.61i)T^{2} \)
5 \( 1 + (5.12 + 1.17i)T + (22.5 + 10.8i)T^{2} \)
7 \( 1 + (6.56 - 3.16i)T + (30.5 - 38.3i)T^{2} \)
11 \( 1 + (0.480 - 4.26i)T + (-117. - 26.9i)T^{2} \)
13 \( 1 + (-2.73 + 2.17i)T + (37.6 - 164. i)T^{2} \)
17 \( 1 + (6.15 + 6.15i)T + 289iT^{2} \)
19 \( 1 + (-5.18 - 14.8i)T + (-282. + 225. i)T^{2} \)
23 \( 1 + (3.58 + 15.7i)T + (-476. + 229. i)T^{2} \)
31 \( 1 + (34.6 - 21.8i)T + (416. - 865. i)T^{2} \)
37 \( 1 + (5.62 + 49.9i)T + (-1.33e3 + 304. i)T^{2} \)
41 \( 1 + (0.940 - 0.940i)T - 1.68e3iT^{2} \)
43 \( 1 + (10.2 - 16.3i)T + (-802. - 1.66e3i)T^{2} \)
47 \( 1 + (59.1 + 6.66i)T + (2.15e3 + 491. i)T^{2} \)
53 \( 1 + (8.89 - 38.9i)T + (-2.53e3 - 1.21e3i)T^{2} \)
59 \( 1 - 6.74T + 3.48e3T^{2} \)
61 \( 1 + (-74.0 - 25.9i)T + (2.90e3 + 2.32e3i)T^{2} \)
67 \( 1 + (-29.6 - 23.6i)T + (998. + 4.37e3i)T^{2} \)
71 \( 1 + (-65.3 + 52.0i)T + (1.12e3 - 4.91e3i)T^{2} \)
73 \( 1 + (38.0 + 23.8i)T + (2.31e3 + 4.80e3i)T^{2} \)
79 \( 1 + (120. - 13.6i)T + (6.08e3 - 1.38e3i)T^{2} \)
83 \( 1 + (-7.07 - 3.40i)T + (4.29e3 + 5.38e3i)T^{2} \)
89 \( 1 + (94.3 - 59.3i)T + (3.43e3 - 7.13e3i)T^{2} \)
97 \( 1 + (13.0 - 4.56i)T + (7.35e3 - 5.86e3i)T^{2} \)
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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

−16.39893189425842696473278772728, −15.91168729542790061802049600618, −14.73763408081321576333734425535, −12.29813590574804372729865809783, −11.09351881550986896700829669709, −9.612823707609008283586474559180, −8.358859343762755575316241919745, −7.00249672070072389800738930879, −5.80355889069658188716128464523, −0.05084524685827240654317412974, 3.46634478136846994510818229595, 6.95912847978972738639899830936, 8.372370253764244689510847521424, 9.845489925486203494800165633379, 11.07048394703905421537377739180, 11.57638043757604854141600071158, 13.03513826772927490015549486029, 15.78458805813984800796246179087, 16.48315914365042584666768014251, 17.42744572115861841452773928672

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