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Properties

Label 2-725-29.9-c1-0-7
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: $\chi_{725} (676, \cdot )$
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

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

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