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Properties

Label 2-29-29.17-c10-0-8
Degree $2$
Conductor $29$
Sign $-0.849 - 0.528i$
Analytic cond. $18.4253$
Root an. cond. $4.29247$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.04 − 4.04i)2-s + (−142. + 142. i)3-s + 991. i·4-s + 351. i·5-s + 1.15e3i·6-s + 2.77e4·7-s + (8.14e3 + 8.14e3i)8-s + 1.82e4i·9-s + (1.42e3 + 1.42e3i)10-s + (−5.40e4 + 5.40e4i)11-s + (−1.41e5 − 1.41e5i)12-s − 1.14e4i·13-s + (1.12e5 − 1.12e5i)14-s + (−5.01e4 − 5.01e4i)15-s − 9.49e5·16-s + (−4.56e5 + 4.56e5i)17-s + ⋯
L(s)  = 1  + (0.126 − 0.126i)2-s + (−0.587 + 0.587i)3-s + 0.968i·4-s + 0.112i·5-s + 0.148i·6-s + 1.64·7-s + (0.248 + 0.248i)8-s + 0.309i·9-s + (0.0142 + 0.0142i)10-s + (−0.335 + 0.335i)11-s + (−0.568 − 0.568i)12-s − 0.0308i·13-s + (0.208 − 0.208i)14-s + (−0.0660 − 0.0660i)15-s − 0.905·16-s + (−0.321 + 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.849 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.849 - 0.528i$
Analytic conductor: \(18.4253\)
Root analytic conductor: \(4.29247\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :5),\ -0.849 - 0.528i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.407520 + 1.42700i\)
\(L(\frac12)\) \(\approx\) \(0.407520 + 1.42700i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (1.91e7 + 7.34e6i)T \)
good2 \( 1 + (-4.04 + 4.04i)T - 1.02e3iT^{2} \)
3 \( 1 + (142. - 142. i)T - 5.90e4iT^{2} \)
5 \( 1 - 351. iT - 9.76e6T^{2} \)
7 \( 1 - 2.77e4T + 2.82e8T^{2} \)
11 \( 1 + (5.40e4 - 5.40e4i)T - 2.59e10iT^{2} \)
13 \( 1 + 1.14e4iT - 1.37e11T^{2} \)
17 \( 1 + (4.56e5 - 4.56e5i)T - 2.01e12iT^{2} \)
19 \( 1 + (1.18e6 - 1.18e6i)T - 6.13e12iT^{2} \)
23 \( 1 + 2.44e5T + 4.14e13T^{2} \)
31 \( 1 + (2.32e7 - 2.32e7i)T - 8.19e14iT^{2} \)
37 \( 1 + (-5.79e6 - 5.79e6i)T + 4.80e15iT^{2} \)
41 \( 1 + (5.34e7 + 5.34e7i)T + 1.34e16iT^{2} \)
43 \( 1 + (-8.97e7 + 8.97e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (-9.78e7 - 9.78e7i)T + 5.25e16iT^{2} \)
53 \( 1 - 3.30e8T + 1.74e17T^{2} \)
59 \( 1 + 7.97e8T + 5.11e17T^{2} \)
61 \( 1 + (-2.47e8 + 2.47e8i)T - 7.13e17iT^{2} \)
67 \( 1 + 7.15e8iT - 1.82e18T^{2} \)
71 \( 1 - 2.37e9iT - 3.25e18T^{2} \)
73 \( 1 + (-1.07e9 - 1.07e9i)T + 4.29e18iT^{2} \)
79 \( 1 + (2.79e9 - 2.79e9i)T - 9.46e18iT^{2} \)
83 \( 1 - 1.12e9T + 1.55e19T^{2} \)
89 \( 1 + (6.56e9 - 6.56e9i)T - 3.11e19iT^{2} \)
97 \( 1 + (-5.86e9 - 5.86e9i)T + 7.37e19iT^{2} \)
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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

−15.37955175007753427853952018335, −14.10430452995013191961527763800, −12.63112671229918106960240478629, −11.36618361633738723452340654914, −10.63994037988952644605145356267, −8.571839093254800148217301697757, −7.47436977452473466962755761812, −5.21757228815784332965766716673, −4.16437803818411843990683140658, −2.06074939836857952694814959563, 0.57504857879336579326146953701, 1.78491503609346270527540527213, 4.72011072363140723743367066891, 5.84139999330448083469766415761, 7.29756515662928978040267717501, 8.963986598543189265015210665851, 10.81009370632788540362355354107, 11.53999191517180162908774719083, 13.11314527818775119687062853929, 14.43458395232494317451813184665

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