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Properties

Label 2-725-29.13-c1-0-41
Degree $2$
Conductor $725$
Sign $-0.830 - 0.557i$
Analytic cond. $5.78915$
Root an. cond. $2.40606$
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 + (1.03 − 0.496i)6-s + (−0.0468 + 0.0587i)7-s + (−7.03 + 1.60i)8-s + (−0.624 − 2.73i)9-s + (−3.68 − 0.840i)11-s − 2.09i·12-s + (−0.196 + 0.858i)13-s + (0.0848 + 0.176i)14-s + (−2.05 + 9.00i)16-s − 3.94i·17-s + (−7.12 − 1.62i)18-s + (−0.557 + 0.444i)19-s + ⋯
L(s)  = 1  + (0.798 − 1.65i)2-s + (0.198 + 0.158i)3-s + (−1.48 − 1.86i)4-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 + (−1.11 − 0.253i)11-s − 0.605i·12-s + (−0.0543 + 0.238i)13-s + (0.0226 + 0.0470i)14-s + (−0.513 + 2.25i)16-s − 0.955i·17-s + (−1.67 − 0.383i)18-s + (−0.127 + 0.102i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $-0.830 - 0.557i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (651, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ -0.830 - 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473704 + 1.55622i\)
\(L(\frac12)\) \(\approx\) \(0.473704 + 1.55622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
29 \( 1 + (0.719 - 5.33i)T \)
good2 \( 1 + (-1.12 + 2.34i)T + (-1.24 - 1.56i)T^{2} \)
3 \( 1 + (-0.343 - 0.273i)T + (0.667 + 2.92i)T^{2} \)
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

−10.19897366320154262612361893893, −9.301276666818997356561055605266, −8.675742087163959276788758898679, −7.20327886397307867858723113130, −5.83728599609466060224947316808, −5.07267384160334049493502164053, −4.02957163291062449250166554689, −3.12813299861121373478931036295, −2.29302077639983429331134194815, −0.62035206036719279279930515431, 2.53317169754766535255561099001, 3.84372377410720659131961308381, 4.94782652462062223784630729539, 5.49495103951160012923431419758, 6.51870756748028433984811724457, 7.39901479934799868791071168408, 8.130594929598978845297378618696, 8.548503457699600381602016311575, 9.925516102196709735579739873078, 10.85006793788455300827326889917

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