Properties

Degree 2
Conductor 29
Sign $0.904 + 0.425i$
Motivic weight 2
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.909 − 0.909i)2-s + (0.442 − 0.442i)3-s + 2.34i·4-s − 4.16i·5-s − 0.805i·6-s − 9.68·7-s + (5.77 + 5.77i)8-s + 8.60i·9-s + (−3.78 − 3.78i)10-s + (−0.334 + 0.334i)11-s + (1.03 + 1.03i)12-s − 12.2i·13-s + (−8.81 + 8.81i)14-s + (−1.84 − 1.84i)15-s + 1.11·16-s + (6.80 − 6.80i)17-s + ⋯
L(s)  = 1  + (0.454 − 0.454i)2-s + (0.147 − 0.147i)3-s + 0.586i·4-s − 0.832i·5-s − 0.134i·6-s − 1.38·7-s + (0.721 + 0.721i)8-s + 0.956i·9-s + (−0.378 − 0.378i)10-s + (−0.0304 + 0.0304i)11-s + (0.0865 + 0.0865i)12-s − 0.940i·13-s + (−0.629 + 0.629i)14-s + (−0.122 − 0.122i)15-s + 0.0698·16-s + (0.400 − 0.400i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.904 + 0.425i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.904 + 0.425i$
motivic weight  =  \(2\)
character  :  $\chi_{29} (17, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 29,\ (\ :1),\ 0.904 + 0.425i)\)
\(L(\frac{3}{2})\)  \(\approx\)  \(1.12415 - 0.251068i\)
\(L(\frac12)\)  \(\approx\)  \(1.12415 - 0.251068i\)
\(L(2)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \neq 29$,\(F_p(T)\) is a polynomial of degree 2. If $p = 29$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad29 \( 1 + (-28.1 - 7.15i)T \)
good2 \( 1 + (-0.909 + 0.909i)T - 4iT^{2} \)
3 \( 1 + (-0.442 + 0.442i)T - 9iT^{2} \)
5 \( 1 + 4.16iT - 25T^{2} \)
7 \( 1 + 9.68T + 49T^{2} \)
11 \( 1 + (0.334 - 0.334i)T - 121iT^{2} \)
13 \( 1 + 12.2iT - 169T^{2} \)
17 \( 1 + (-6.80 + 6.80i)T - 289iT^{2} \)
19 \( 1 + (-14.6 + 14.6i)T - 361iT^{2} \)
23 \( 1 + 10.0T + 529T^{2} \)
31 \( 1 + (37.3 - 37.3i)T - 961iT^{2} \)
37 \( 1 + (45.0 + 45.0i)T + 1.36e3iT^{2} \)
41 \( 1 + (-22.8 - 22.8i)T + 1.68e3iT^{2} \)
43 \( 1 + (17.5 - 17.5i)T - 1.84e3iT^{2} \)
47 \( 1 + (-2.20 - 2.20i)T + 2.20e3iT^{2} \)
53 \( 1 - 90.1T + 2.80e3T^{2} \)
59 \( 1 + 90.4T + 3.48e3T^{2} \)
61 \( 1 + (29.4 - 29.4i)T - 3.72e3iT^{2} \)
67 \( 1 - 31.5iT - 4.48e3T^{2} \)
71 \( 1 + 99.8iT - 5.04e3T^{2} \)
73 \( 1 + (3.96 + 3.96i)T + 5.32e3iT^{2} \)
79 \( 1 + (40.3 - 40.3i)T - 6.24e3iT^{2} \)
83 \( 1 - 137.T + 6.88e3T^{2} \)
89 \( 1 + (-83.1 + 83.1i)T - 7.92e3iT^{2} \)
97 \( 1 + (79.9 + 79.9i)T + 9.40e3iT^{2} \)
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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

−16.51646880103747692535453490435, −16.01868181820487541463993504476, −13.84994696833320518615562394800, −12.96424428251964812050542134412, −12.26516815167372795761522739183, −10.54336023713537912152418508851, −8.888286674034197731583425219879, −7.42575825059943410272127857067, −5.11422401381085990128893379332, −3.09957726086535191363110545545, 3.64788929232417560239361487827, 6.05667775407692816830534473862, 6.93979564666978871367798565554, 9.434726193625607864928453284584, 10.33678181107372981521418134657, 12.15498605986684551748281788420, 13.66482492402327738565023043578, 14.62615887862688524863274881452, 15.58867123480813835903575747945, 16.60221065080574347138383100082

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