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Properties

Label 2-29-29.17-c6-0-8
Degree $2$
Conductor $29$
Sign $0.939 - 0.342i$
Analytic cond. $6.67156$
Root an. cond. $2.58293$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.260 − 0.260i)2-s + (29.0 − 29.0i)3-s + 63.8i·4-s + 194. i·5-s − 15.1i·6-s + 282.·7-s + (33.3 + 33.3i)8-s − 959. i·9-s + (50.6 + 50.6i)10-s + (664. − 664. i)11-s + (1.85e3 + 1.85e3i)12-s + 386. i·13-s + (73.7 − 73.7i)14-s + (5.64e3 + 5.64e3i)15-s − 4.06e3·16-s + (−1.66e3 + 1.66e3i)17-s + ⋯
L(s)  = 1  + (0.0325 − 0.0325i)2-s + (1.07 − 1.07i)3-s + 0.997i·4-s + 1.55i·5-s − 0.0701i·6-s + 0.824·7-s + (0.0651 + 0.0651i)8-s − 1.31i·9-s + (0.0506 + 0.0506i)10-s + (0.499 − 0.499i)11-s + (1.07 + 1.07i)12-s + 0.175i·13-s + (0.0268 − 0.0268i)14-s + (1.67 + 1.67i)15-s − 0.993·16-s + (−0.339 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(6.67156\)
Root analytic conductor: \(2.58293\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :3),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.35181 + 0.415231i\)
\(L(\frac12)\) \(\approx\) \(2.35181 + 0.415231i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-2.09e4 + 1.25e4i)T \)
good2 \( 1 + (-0.260 + 0.260i)T - 64iT^{2} \)
3 \( 1 + (-29.0 + 29.0i)T - 729iT^{2} \)
5 \( 1 - 194. iT - 1.56e4T^{2} \)
7 \( 1 - 282.T + 1.17e5T^{2} \)
11 \( 1 + (-664. + 664. i)T - 1.77e6iT^{2} \)
13 \( 1 - 386. iT - 4.82e6T^{2} \)
17 \( 1 + (1.66e3 - 1.66e3i)T - 2.41e7iT^{2} \)
19 \( 1 + (-8.74e3 + 8.74e3i)T - 4.70e7iT^{2} \)
23 \( 1 + 1.40e4T + 1.48e8T^{2} \)
31 \( 1 + (2.59e4 - 2.59e4i)T - 8.87e8iT^{2} \)
37 \( 1 + (4.39e4 + 4.39e4i)T + 2.56e9iT^{2} \)
41 \( 1 + (6.30e4 + 6.30e4i)T + 4.75e9iT^{2} \)
43 \( 1 + (-5.59e4 + 5.59e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-8.39e4 - 8.39e4i)T + 1.07e10iT^{2} \)
53 \( 1 - 8.03e4T + 2.21e10T^{2} \)
59 \( 1 + 1.99e5T + 4.21e10T^{2} \)
61 \( 1 + (-1.11e5 + 1.11e5i)T - 5.15e10iT^{2} \)
67 \( 1 + 4.60e5iT - 9.04e10T^{2} \)
71 \( 1 - 5.09e5iT - 1.28e11T^{2} \)
73 \( 1 + (-5.89e4 - 5.89e4i)T + 1.51e11iT^{2} \)
79 \( 1 + (-5.02e5 + 5.02e5i)T - 2.43e11iT^{2} \)
83 \( 1 + 1.84e5T + 3.26e11T^{2} \)
89 \( 1 + (5.75e5 - 5.75e5i)T - 4.96e11iT^{2} \)
97 \( 1 + (4.94e5 + 4.94e5i)T + 8.32e11iT^{2} \)
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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

−15.60406522243600173106351293145, −14.07850607638069080946832252795, −13.88475689397664456589247725247, −12.17504787973237535888191034850, −11.04910240967204802358416678486, −8.829133861905456791182519390705, −7.63573801121010937634631981123, −6.83675397211705339137215015913, −3.45873605971541079331345354374, −2.24177197775673250078988855796, 1.51510155848825945041260615632, 4.26715095234091628672386178006, 5.29036764924478558670938204585, 8.171052746362072532641511292777, 9.260862847323805258861739329221, 10.09146907179066845020288317945, 11.91598751661799180940999043985, 13.73676202501263150568812968799, 14.56671348029836973668924448821, 15.63045503119217573320856653343

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