Properties

Degree 2
Conductor 29
Sign $-0.806 + 0.590i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.68 − 0.190i)2-s + (−3.78 − 2.37i)3-s + (−1.08 − 0.248i)4-s + (0.141 − 0.113i)5-s + (5.93 + 4.73i)6-s + (−1.55 − 6.79i)7-s + (8.20 + 2.86i)8-s + (4.76 + 9.90i)9-s + (−0.260 + 0.163i)10-s + (−12.2 + 4.28i)11-s + (3.52 + 3.52i)12-s + (9.66 − 20.0i)13-s + (1.32 + 11.7i)14-s + (−0.805 + 0.0907i)15-s + (−9.27 − 4.46i)16-s + (−5.90 + 5.90i)17-s + ⋯
L(s)  = 1  + (−0.843 − 0.0950i)2-s + (−1.26 − 0.792i)3-s + (−0.271 − 0.0620i)4-s + (0.0283 − 0.0226i)5-s + (0.989 + 0.789i)6-s + (−0.221 − 0.970i)7-s + (1.02 + 0.358i)8-s + (0.529 + 1.10i)9-s + (−0.0260 + 0.0163i)10-s + (−1.11 + 0.389i)11-s + (0.293 + 0.293i)12-s + (0.743 − 1.54i)13-s + (0.0946 + 0.840i)14-s + (−0.0536 + 0.00605i)15-s + (−0.579 − 0.279i)16-s + (−0.347 + 0.347i)17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.590i)\, \overline{\Lambda}(3-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.806 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $-0.806 + 0.590i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (2, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1),\ -0.806 + 0.590i)$
$L(\frac{3}{2})$  $\approx$  $0.0918967 - 0.280987i$
$L(\frac12)$  $\approx$  $0.0918967 - 0.280987i$
$L(2)$   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 \( 1 + (-28.5 - 4.93i)T \)
good2 \( 1 + (1.68 + 0.190i)T + (3.89 + 0.890i)T^{2} \)
3 \( 1 + (3.78 + 2.37i)T + (3.90 + 8.10i)T^{2} \)
5 \( 1 + (-0.141 + 0.113i)T + (5.56 - 24.3i)T^{2} \)
7 \( 1 + (1.55 + 6.79i)T + (-44.1 + 21.2i)T^{2} \)
11 \( 1 + (12.2 - 4.28i)T + (94.6 - 75.4i)T^{2} \)
13 \( 1 + (-9.66 + 20.0i)T + (-105. - 132. i)T^{2} \)
17 \( 1 + (5.90 - 5.90i)T - 289iT^{2} \)
19 \( 1 + (10.7 + 17.0i)T + (-156. + 325. i)T^{2} \)
23 \( 1 + (-15.0 + 18.8i)T + (-117. - 515. i)T^{2} \)
31 \( 1 + (-35.5 - 4.01i)T + (936. + 213. i)T^{2} \)
37 \( 1 + (23.0 + 8.08i)T + (1.07e3 + 853. i)T^{2} \)
41 \( 1 + (14.1 + 14.1i)T + 1.68e3iT^{2} \)
43 \( 1 + (4.31 + 38.2i)T + (-1.80e3 + 411. i)T^{2} \)
47 \( 1 + (-1.53 - 4.38i)T + (-1.72e3 + 1.37e3i)T^{2} \)
53 \( 1 + (51.0 + 63.9i)T + (-625. + 2.73e3i)T^{2} \)
59 \( 1 - 28.0T + 3.48e3T^{2} \)
61 \( 1 + (3.26 + 2.05i)T + (1.61e3 + 3.35e3i)T^{2} \)
67 \( 1 + (2.88 + 5.98i)T + (-2.79e3 + 3.50e3i)T^{2} \)
71 \( 1 + (15.5 - 32.3i)T + (-3.14e3 - 3.94e3i)T^{2} \)
73 \( 1 + (-103. + 11.6i)T + (5.19e3 - 1.18e3i)T^{2} \)
79 \( 1 + (28.1 - 80.4i)T + (-4.87e3 - 3.89e3i)T^{2} \)
83 \( 1 + (6.13 - 26.8i)T + (-6.20e3 - 2.98e3i)T^{2} \)
89 \( 1 + (-26.8 - 3.02i)T + (7.72e3 + 1.76e3i)T^{2} \)
97 \( 1 + (6.09 - 3.82i)T + (4.08e3 - 8.47e3i)T^{2} \)
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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

−17.12134145935416764514379532630, −15.63696455354957788755515892990, −13.43040438846766280399673870406, −12.81543658869714031325250173830, −10.87285025927974108978696366427, −10.36676472029716187606545770058, −8.254671985057831799196882013461, −6.92404771331809003635787719328, −5.12278031123930835660794851337, −0.59471600967433110349041754503, 4.64978391921649326850679589896, 6.21692410862694585012655780562, 8.456377567230413767980003101729, 9.709366027429817231919714006585, 10.82403455733075773587641559745, 12.02249255179287235568861340251, 13.65982761198351631922743448444, 15.71940913984013885284299438876, 16.26957506983263011671126759274, 17.35742438943790045668889457260

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