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Properties

Label 2-29-29.13-c1-0-0
Degree $2$
Conductor $29$
Sign $-0.0821 - 0.996i$
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  + (−1.12 + 2.34i)2-s + (−0.343 − 0.273i)3-s + (−2.97 − 3.73i)4-s + (2.32 + 1.11i)5-s + (1.03 − 0.496i)6-s + (0.0468 − 0.0587i)7-s + (7.03 − 1.60i)8-s + (−0.624 − 2.73i)9-s + (−5.25 + 4.18i)10-s + (−3.68 − 0.840i)11-s + 2.09i·12-s + (0.196 − 0.858i)13-s + (0.0848 + 0.176i)14-s + (−0.491 − 1.02i)15-s + (−2.05 + 9.00i)16-s + 3.94i·17-s + ⋯
L(s)  = 1  + (−0.798 + 1.65i)2-s + (−0.198 − 0.158i)3-s + (−1.48 − 1.86i)4-s + (1.03 + 0.500i)5-s + (0.420 − 0.202i)6-s + (0.0177 − 0.0222i)7-s + (2.48 − 0.567i)8-s + (−0.208 − 0.912i)9-s + (−1.66 + 1.32i)10-s + (−1.11 − 0.253i)11-s + 0.605i·12-s + (0.0543 − 0.238i)13-s + (0.0226 + 0.0470i)14-s + (−0.127 − 0.263i)15-s + (−0.513 + 2.25i)16-s + 0.955i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0821 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0821 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(29\)
Sign: $-0.0821 - 0.996i$
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: \((2,\ 29,\ (\ :1/2),\ -0.0821 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.329771 + 0.358069i\)
\(L(\frac12)\) \(\approx\) \(0.329771 + 0.358069i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad29 \( 1 + (0.719 - 5.33i)T \)
good2 \( 1 + (1.12 - 2.34i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (0.343 + 0.273i)T + (0.667 + 2.92i)T^{2} \)
5 \( 1 + (-2.32 - 1.11i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-0.0468 + 0.0587i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (3.68 + 0.840i)T + (9.91 + 4.77i)T^{2} \)
13 \( 1 + (-0.196 + 0.858i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 3.94iT - 17T^{2} \)
19 \( 1 + (0.557 - 0.444i)T + (4.22 - 18.5i)T^{2} \)
23 \( 1 + (1.06 - 0.510i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-2.23 + 4.64i)T + (-19.3 - 24.2i)T^{2} \)
37 \( 1 + (-3.01 + 0.687i)T + (33.3 - 16.0i)T^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 + (-3.60 - 7.49i)T + (-26.8 + 33.6i)T^{2} \)
47 \( 1 + (10.3 + 2.35i)T + (42.3 + 20.3i)T^{2} \)
53 \( 1 + (5.00 + 2.41i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 - 9.91T + 59T^{2} \)
61 \( 1 + (-2.78 - 2.22i)T + (13.5 + 59.4i)T^{2} \)
67 \( 1 + (-1.09 - 4.81i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.09 + 4.78i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (3.86 + 8.02i)T + (-45.5 + 57.0i)T^{2} \)
79 \( 1 + (-12.8 + 2.93i)T + (71.1 - 34.2i)T^{2} \)
83 \( 1 + (-10.5 - 13.2i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-2.94 + 6.12i)T + (-55.4 - 69.5i)T^{2} \)
97 \( 1 + (8.92 - 7.11i)T + (21.5 - 94.5i)T^{2} \)
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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

−17.69801949667028702669107464344, −16.45301898965285451612592283932, −15.21847880026371795787382718076, −14.34545758027258990851532096854, −13.08430359340538416507439773911, −10.56258177366848806189191367173, −9.463296604949100096754537495739, −8.047921261237199841767406783348, −6.50583116542003016405616963010, −5.62874404433070015118395725525, 2.33858877236063270087836019945, 4.99381921463705267630500825165, 8.116744255044911175206520282486, 9.524447204125726758034182209413, 10.37969520539638674763785400332, 11.56029144187068567916546439583, 12.98395802729557846023467737189, 13.70721243132609339947521677206, 16.24138594111243637125395801471, 17.34635068652096680478349889078

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