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Properties

Label 2-725-29.16-c1-0-9
Degree $2$
Conductor $725$
Sign $0.441 - 0.897i$
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  + (−0.400 − 0.193i)2-s + (0.777 − 0.974i)3-s + (−1.12 − 1.40i)4-s + (−0.5 + 0.240i)6-s + (−0.222 + 0.279i)7-s + (0.376 + 1.64i)8-s + (0.321 + 1.40i)9-s + (−1.09 + 4.81i)11-s − 2.24·12-s + (−1.25 + 5.51i)13-s + (0.143 − 0.0689i)14-s + (−0.634 + 2.77i)16-s − 4.49·17-s + (0.143 − 0.626i)18-s + (−1.46 − 1.84i)19-s + ⋯
L(s)  = 1  + (−0.283 − 0.136i)2-s + (0.448 − 0.562i)3-s + (−0.561 − 0.704i)4-s + (−0.204 + 0.0983i)6-s + (−0.0841 + 0.105i)7-s + (0.133 + 0.583i)8-s + (0.107 + 0.469i)9-s + (−0.331 + 1.45i)11-s − 0.648·12-s + (−0.348 + 1.52i)13-s + (0.0382 − 0.0184i)14-s + (−0.158 + 0.694i)16-s − 1.08·17-s + (0.0337 − 0.147i)18-s + (−0.337 − 0.422i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.441 - 0.897i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.441 - 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.712197 + 0.443444i\)
\(L(\frac12)\) \(\approx\) \(0.712197 + 0.443444i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.09 + 1.73i)T \)
good2 \( 1 + (0.400 + 0.193i)T + (1.24 + 1.56i)T^{2} \)
3 \( 1 + (-0.777 + 0.974i)T + (-0.667 - 2.92i)T^{2} \)
7 \( 1 + (0.222 - 0.279i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (1.09 - 4.81i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (1.25 - 5.51i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + (1.46 + 1.84i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (-2.06 + 0.996i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (-6.02 - 2.90i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (1.09 + 4.81i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 - 3.10T + 41T^{2} \)
43 \( 1 + (3.06 - 1.47i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (1.43 - 6.28i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (4.22 + 2.03i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 12.4T + 59T^{2} \)
61 \( 1 + (1.02 - 1.28i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (0.516 + 2.26i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-1.63 + 7.15i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (-5.06 + 2.44i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (-1.03 - 4.54i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (2.77 + 3.48i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (-5.11 - 2.46i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-0.112 - 0.141i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.51416633736959254888028329655, −9.406543439418259132688965456649, −9.126088205769461381953077609879, −7.999601231202406618374823756448, −7.12046156210336464619060733136, −6.34224305112678227831992593421, −4.78764279706192559590401521163, −4.49155208272479437146272177747, −2.38249258866783396995711421124, −1.72743695889751321779725478777, 0.44993679197153143187270202057, 2.94622801998361833055124395656, 3.53406359627434712043597100102, 4.59760755327555248945841427620, 5.74050609503776823558459682178, 6.87340184518462517433365844459, 8.052732773820569399545587092546, 8.466818853159313866630971153145, 9.282154863253267884231902946842, 10.09223566228414211444630352654

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