Properties

Label 2-29-29.15-c2-0-1
Degree $2$
Conductor $29$
Sign $0.999 - 0.0174i$
Analytic cond. $0.790192$
Root an. cond. $0.888927$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.56 + 0.401i)2-s + (1.78 − 1.12i)3-s + (8.62 − 1.96i)4-s + (5.24 + 4.17i)5-s + (−5.90 + 4.70i)6-s + (2.11 − 9.28i)7-s + (−16.3 + 5.73i)8-s + (−1.97 + 4.11i)9-s + (−20.3 − 12.7i)10-s + (−0.700 − 0.245i)11-s + (13.1 − 13.1i)12-s + (3.39 + 7.04i)13-s + (−3.82 + 33.9i)14-s + (14.0 + 1.58i)15-s + (24.1 − 11.6i)16-s + (−17.4 − 17.4i)17-s + ⋯
L(s)  = 1  + (−1.78 + 0.200i)2-s + (0.594 − 0.373i)3-s + (2.15 − 0.492i)4-s + (1.04 + 0.835i)5-s + (−0.983 + 0.784i)6-s + (0.302 − 1.32i)7-s + (−2.04 + 0.716i)8-s + (−0.219 + 0.456i)9-s + (−2.03 − 1.27i)10-s + (−0.0637 − 0.0222i)11-s + (1.09 − 1.09i)12-s + (0.261 + 0.542i)13-s + (−0.273 + 2.42i)14-s + (0.935 + 0.105i)15-s + (1.51 − 0.727i)16-s + (−1.02 − 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0174i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0174i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.999 - 0.0174i$
Analytic conductor: \(0.790192\)
Root analytic conductor: \(0.888927\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1),\ 0.999 - 0.0174i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.623147 + 0.00544588i\)
\(L(\frac12)\) \(\approx\) \(0.623147 + 0.00544588i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (19.7 - 21.2i)T \)
good2 \( 1 + (3.56 - 0.401i)T + (3.89 - 0.890i)T^{2} \)
3 \( 1 + (-1.78 + 1.12i)T + (3.90 - 8.10i)T^{2} \)
5 \( 1 + (-5.24 - 4.17i)T + (5.56 + 24.3i)T^{2} \)
7 \( 1 + (-2.11 + 9.28i)T + (-44.1 - 21.2i)T^{2} \)
11 \( 1 + (0.700 + 0.245i)T + (94.6 + 75.4i)T^{2} \)
13 \( 1 + (-3.39 - 7.04i)T + (-105. + 132. i)T^{2} \)
17 \( 1 + (17.4 + 17.4i)T + 289iT^{2} \)
19 \( 1 + (7.96 - 12.6i)T + (-156. - 325. i)T^{2} \)
23 \( 1 + (11.9 + 15.0i)T + (-117. + 515. i)T^{2} \)
31 \( 1 + (-0.327 + 0.0369i)T + (936. - 213. i)T^{2} \)
37 \( 1 + (15.2 - 5.34i)T + (1.07e3 - 853. i)T^{2} \)
41 \( 1 + (-28.0 + 28.0i)T - 1.68e3iT^{2} \)
43 \( 1 + (-0.679 + 6.02i)T + (-1.80e3 - 411. i)T^{2} \)
47 \( 1 + (-1.43 + 4.10i)T + (-1.72e3 - 1.37e3i)T^{2} \)
53 \( 1 + (37.6 - 47.1i)T + (-625. - 2.73e3i)T^{2} \)
59 \( 1 - 91.1T + 3.48e3T^{2} \)
61 \( 1 + (-6.43 + 4.04i)T + (1.61e3 - 3.35e3i)T^{2} \)
67 \( 1 + (-29.8 + 62.0i)T + (-2.79e3 - 3.50e3i)T^{2} \)
71 \( 1 + (38.1 + 79.2i)T + (-3.14e3 + 3.94e3i)T^{2} \)
73 \( 1 + (29.1 + 3.28i)T + (5.19e3 + 1.18e3i)T^{2} \)
79 \( 1 + (-7.28 - 20.8i)T + (-4.87e3 + 3.89e3i)T^{2} \)
83 \( 1 + (-9.19 - 40.2i)T + (-6.20e3 + 2.98e3i)T^{2} \)
89 \( 1 + (-98.0 + 11.0i)T + (7.72e3 - 1.76e3i)T^{2} \)
97 \( 1 + (-136. - 86.0i)T + (4.08e3 + 8.47e3i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.20381420185940539141020432792, −16.25641486207767262262849749484, −14.43066149536923724815595972320, −13.63106037057178340601902540659, −11.02591209877167334834463414520, −10.33181459063392946598500960859, −9.004828155093114908088000586703, −7.60532482196887026663776969558, −6.63320842727198425524264247209, −2.09147568901355725236710768500, 2.17403251881833238533724983390, 5.99760421440243241711211321872, 8.403278286763391354245687720317, 8.972834492986334023686666700909, 9.876193411912551826500286347404, 11.46016721054536833326264770882, 12.96155709527904930107817240449, 15.01496316256726878822835922114, 15.90876703702229667732650243733, 17.53534610553738230695584125403

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