Properties

Degree 2
Conductor 29
Sign $0.976 - 0.214i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.400 + 0.193i)2-s + (−0.777 + 0.974i)3-s + (−1.12 − 1.40i)4-s + (−0.623 − 0.300i)5-s + (−0.5 + 0.240i)6-s + (0.222 − 0.279i)7-s + (−0.376 − 1.64i)8-s + (0.321 + 1.40i)9-s + (−0.192 − 0.240i)10-s + (−1.09 + 4.81i)11-s + 2.24·12-s + (1.25 − 5.51i)13-s + (0.143 − 0.0689i)14-s + (0.777 − 0.374i)15-s + (−0.634 + 2.77i)16-s + 4.49·17-s + ⋯
L(s)  = 1  + (0.283 + 0.136i)2-s + (−0.448 + 0.562i)3-s + (−0.561 − 0.704i)4-s + (−0.278 − 0.134i)5-s + (−0.204 + 0.0983i)6-s + (0.0841 − 0.105i)7-s + (−0.133 − 0.583i)8-s + (0.107 + 0.469i)9-s + (−0.0607 − 0.0761i)10-s + (−0.331 + 1.45i)11-s + 0.648·12-s + (0.348 − 1.52i)13-s + (0.0382 − 0.0184i)14-s + (0.200 − 0.0966i)15-s + (−0.158 + 0.694i)16-s + 1.08·17-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(29\)
\( \varepsilon \)  =  $0.976 - 0.214i$
motivic weight  =  \(1\)
character  :  $\chi_{29} (16, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 29,\ (\ :1/2),\ 0.976 - 0.214i)$
$L(1)$  $\approx$  $0.635745 + 0.0689952i$
$L(\frac12)$  $\approx$  $0.635745 + 0.0689952i$
$L(\frac{3}{2})$   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 \( 1 + (5.09 + 1.73i)T \)
good2 \( 1 + (-0.400 - 0.193i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (0.777 - 0.974i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (0.623 + 0.300i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-0.222 + 0.279i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (1.09 - 4.81i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-1.25 + 5.51i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 4.49T + 17T^{2} \)
19 \( 1 + (1.46 + 1.84i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (2.06 - 0.996i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-6.02 - 2.90i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (-1.09 - 4.81i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 - 3.10T + 41T^{2} \)
43 \( 1 + (-3.06 + 1.47i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.43 + 6.28i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (-4.22 - 2.03i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + (1.02 - 1.28i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-0.516 - 2.26i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.63 + 7.15i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (5.06 - 2.44i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (-1.03 - 4.54i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (-2.77 - 3.48i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-5.11 - 2.46i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (0.112 + 0.141i)T + (-21.5 + 94.5i)T^{2} \)
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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

−17.25737412043048777024071935888, −15.70315092209086603554001914016, −15.10250980532061201121595665962, −13.60184586560095550438219539101, −12.39360401601599654544105072220, −10.56142074247679254657255440394, −9.850540473922797701672075212363, −7.83506387503764677641644822333, −5.63591874046606001267864523225, −4.45939968424702078647269062605, 3.76461178074972908672241363844, 5.96165855796953113502760425775, 7.74090712307790665573723381081, 9.136575573695839191221125418177, 11.32356232577992504312304893363, 12.13358214107421180852385838280, 13.36966913753902926624511352925, 14.40364204832305966722543618039, 16.24860856719243446714544977752, 17.13173060305881366439228969749

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