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Properties

Label 2-29-29.5-c1-0-0
Degree $2$
Conductor $29$
Sign $0.868 - 0.496i$
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  + (−0.909 + 0.725i)2-s + (0.960 − 0.219i)3-s + (−0.144 + 0.631i)4-s + (−1.18 − 1.48i)5-s + (−0.714 + 0.895i)6-s + (−0.339 − 1.48i)7-s + (−1.33 − 2.77i)8-s + (−1.82 + 0.880i)9-s + (2.15 + 0.491i)10-s + (−0.344 + 0.716i)11-s + 0.637i·12-s + (5.80 + 2.79i)13-s + (1.38 + 1.10i)14-s + (−1.46 − 1.16i)15-s + (2.06 + 0.992i)16-s + 4.03i·17-s + ⋯
L(s)  = 1  + (−0.642 + 0.512i)2-s + (0.554 − 0.126i)3-s + (−0.0720 + 0.315i)4-s + (−0.529 − 0.664i)5-s + (−0.291 + 0.365i)6-s + (−0.128 − 0.562i)7-s + (−0.472 − 0.980i)8-s + (−0.609 + 0.293i)9-s + (0.681 + 0.155i)10-s + (−0.103 + 0.215i)11-s + 0.184i·12-s + (1.61 + 0.775i)13-s + (0.370 + 0.295i)14-s + (−0.377 − 0.301i)15-s + (0.515 + 0.248i)16-s + 0.977i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.868 - 0.496i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $0.868 - 0.496i$
Analytic conductor: \(0.231566\)
Root analytic conductor: \(0.481213\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{29} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 29,\ (\ :1/2),\ 0.868 - 0.496i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.531901 + 0.141258i\)
\(L(\frac12)\) \(\approx\) \(0.531901 + 0.141258i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (-0.825 - 5.32i)T \)
good2 \( 1 + (0.909 - 0.725i)T + (0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.960 + 0.219i)T + (2.70 - 1.30i)T^{2} \)
5 \( 1 + (1.18 + 1.48i)T + (-1.11 + 4.87i)T^{2} \)
7 \( 1 + (0.339 + 1.48i)T + (-6.30 + 3.03i)T^{2} \)
11 \( 1 + (0.344 - 0.716i)T + (-6.85 - 8.60i)T^{2} \)
13 \( 1 + (-5.80 - 2.79i)T + (8.10 + 10.1i)T^{2} \)
17 \( 1 - 4.03iT - 17T^{2} \)
19 \( 1 + (5.82 + 1.33i)T + (17.1 + 8.24i)T^{2} \)
23 \( 1 + (-3.63 + 4.55i)T + (-5.11 - 22.4i)T^{2} \)
31 \( 1 + (-0.493 + 0.393i)T + (6.89 - 30.2i)T^{2} \)
37 \( 1 + (1.09 + 2.28i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 2.49iT - 41T^{2} \)
43 \( 1 + (6.40 + 5.11i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-2.92 + 6.08i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (0.429 + 0.539i)T + (-11.7 + 51.6i)T^{2} \)
59 \( 1 - 2.67T + 59T^{2} \)
61 \( 1 + (7.97 - 1.82i)T + (54.9 - 26.4i)T^{2} \)
67 \( 1 + (-3.56 + 1.71i)T + (41.7 - 52.3i)T^{2} \)
71 \( 1 + (2.81 + 1.35i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (-0.715 - 0.570i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (-5.58 - 11.6i)T + (-49.2 + 61.7i)T^{2} \)
83 \( 1 + (2.80 - 12.3i)T + (-74.7 - 36.0i)T^{2} \)
89 \( 1 + (1.43 - 1.14i)T + (19.8 - 86.7i)T^{2} \)
97 \( 1 + (0.117 + 0.0268i)T + (87.3 + 42.0i)T^{2} \)
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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

−16.96683526588237443878062494175, −16.44598661447552413106385461505, −15.13375627977013246766020365513, −13.57183283059948194646518721976, −12.54161854412030257467253134683, −10.80436619422367268361323359685, −8.718986923675579847525050955243, −8.404170587093459390918590619391, −6.70990106790263234729638205098, −3.95733245259119593742700930144, 3.07956168117191385141463103997, 5.95756030164411865972027242283, 8.226297919743338547195838987203, 9.210057392723494619948530262420, 10.74787669059536159367688583079, 11.60925037981982128894431044817, 13.55297182176881227599604595170, 14.87242711098006932739383292037, 15.57403050739905707103812046379, 17.49189071676708597016785800991

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