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Properties

Label 2-29-29.13-c1-0-1
Degree $2$
Conductor $29$
Sign $0.986 + 0.164i$
Analytic cond. $0.231566$
Root an. cond. $0.481213$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0741 − 0.154i)2-s + (−0.879 − 0.701i)3-s + (1.22 + 1.54i)4-s + (−2.54 − 1.22i)5-s + (−0.173 + 0.0834i)6-s + (−1.82 + 2.28i)7-s + (0.662 − 0.151i)8-s + (−0.386 − 1.69i)9-s + (−0.378 + 0.301i)10-s + (3.89 + 0.888i)11-s − 2.21i·12-s + (0.625 − 2.74i)13-s + (0.217 + 0.450i)14-s + (1.37 + 2.86i)15-s + (−0.851 + 3.72i)16-s + 0.482i·17-s + ⋯
L(s)  = 1  + (0.0524 − 0.108i)2-s + (−0.507 − 0.404i)3-s + (0.614 + 0.770i)4-s + (−1.13 − 0.548i)5-s + (−0.0707 + 0.0340i)6-s + (−0.689 + 0.864i)7-s + (0.234 − 0.0534i)8-s + (−0.128 − 0.564i)9-s + (−0.119 + 0.0953i)10-s + (1.17 + 0.267i)11-s − 0.639i·12-s + (0.173 − 0.760i)13-s + (0.0580 + 0.120i)14-s + (0.356 + 0.739i)15-s + (−0.212 + 0.932i)16-s + 0.117i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(0.231566\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1/2),\ 0.986 + 0.164i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612546 - 0.0508684i\)
\(L(\frac12)\) \(\approx\) \(0.612546 - 0.0508684i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (4.99 + 2.00i)T \)
good2 \( 1 + (-0.0741 + 0.154i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (0.879 + 0.701i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (2.54 + 1.22i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (1.82 - 2.28i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (-3.89 - 0.888i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.625 + 2.74i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 0.482iT - 17T^{2} \)
19 \( 1 + (-2.38 + 1.90i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (4.96 - 2.39i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (1.67 - 3.47i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-11.2 + 2.56i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 - 5.10iT - 41T^{2} \)
43 \( 1 + (3.56 + 7.40i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (-2.32 - 0.531i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (-0.401 - 0.193i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 1.24T + 59T^{2} \)
61 \( 1 + (6.71 + 5.35i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (0.210 + 0.921i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.33 + 5.82i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-0.209 - 0.435i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (-1.80 + 0.411i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-2.71 - 3.40i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-6.75 + 14.0i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (-1.88 + 1.50i)T + (21.5 - 94.5i)T^{2} \)
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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

−17.05033589777420874404518799515, −16.00260433388856971168193768160, −15.15341468130861380121988456841, −12.86014555705453624588421110649, −12.07050477197027516797568639871, −11.50840114469240598698334959006, −9.135913226285766428805878181276, −7.66701423508775556807818324872, −6.20632046541944219261412287523, −3.64588921002318162183508342745, 4.04433579375012822567938171473, 6.23380234083278213348474027600, 7.47509254722675608135405081561, 9.815207748564713566227809405726, 11.04767643623673313591553656124, 11.70359812828073884696619245259, 13.87369722205306713180644618555, 14.92651124464482794326059335616, 16.34615609945955921773230442333, 16.52644638739892853264572663897

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