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Properties

Label 2-725-145.24-c1-0-29
Degree $2$
Conductor $725$
Sign $0.869 + 0.494i$
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  + (−1.40 + 1.12i)2-s + (0.433 − 0.0990i)3-s + (0.277 − 1.21i)4-s + (−0.500 + 0.626i)6-s + (3.94 − 0.900i)7-s + (−0.588 − 1.22i)8-s + (−2.52 + 1.21i)9-s + (−2.62 − 1.26i)11-s − 0.554i·12-s + (2.25 − 4.67i)13-s + (−4.54 + 5.70i)14-s + (4.44 + 2.14i)16-s − 1.10i·17-s + (2.19 − 4.54i)18-s + (0.455 − 1.99i)19-s + ⋯
L(s)  = 1  + (−0.996 + 0.794i)2-s + (0.250 − 0.0571i)3-s + (0.138 − 0.607i)4-s + (−0.204 + 0.255i)6-s + (1.49 − 0.340i)7-s + (−0.208 − 0.432i)8-s + (−0.841 + 0.405i)9-s + (−0.791 − 0.380i)11-s − 0.160i·12-s + (0.624 − 1.29i)13-s + (−1.21 + 1.52i)14-s + (1.11 + 0.535i)16-s − 0.269i·17-s + (0.516 − 1.07i)18-s + (0.104 − 0.458i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (24, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.802598 - 0.212401i\)
\(L(\frac12)\) \(\approx\) \(0.802598 - 0.212401i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.38 + 3.12i)T \)
good2 \( 1 + (1.40 - 1.12i)T + (0.445 - 1.94i)T^{2} \)
3 \( 1 + (-0.433 + 0.0990i)T + (2.70 - 1.30i)T^{2} \)
7 \( 1 + (-3.94 + 0.900i)T + (6.30 - 3.03i)T^{2} \)
11 \( 1 + (2.62 + 1.26i)T + (6.85 + 8.60i)T^{2} \)
13 \( 1 + (-2.25 + 4.67i)T + (-8.10 - 10.1i)T^{2} \)
17 \( 1 + 1.10iT - 17T^{2} \)
19 \( 1 + (-0.455 + 1.99i)T + (-17.1 - 8.24i)T^{2} \)
23 \( 1 + (3.23 + 2.57i)T + (5.11 + 22.4i)T^{2} \)
31 \( 1 + (3.96 + 4.97i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (1.26 + 2.62i)T + (-23.0 + 28.9i)T^{2} \)
41 \( 1 - 0.396T + 41T^{2} \)
43 \( 1 + (-4.48 - 3.57i)T + (9.56 + 41.9i)T^{2} \)
47 \( 1 + (-3.38 + 7.02i)T + (-29.3 - 36.7i)T^{2} \)
53 \( 1 + (-3.40 + 2.71i)T + (11.7 - 51.6i)T^{2} \)
59 \( 1 - 9.10T + 59T^{2} \)
61 \( 1 + (-1.34 - 5.89i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-0.162 - 0.337i)T + (-41.7 + 52.3i)T^{2} \)
71 \( 1 + (-10.2 - 4.94i)T + (44.2 + 55.5i)T^{2} \)
73 \( 1 + (6.99 + 5.57i)T + (16.2 + 71.1i)T^{2} \)
79 \( 1 + (0.535 - 0.257i)T + (49.2 - 61.7i)T^{2} \)
83 \( 1 + (-9.19 - 2.09i)T + (74.7 + 36.0i)T^{2} \)
89 \( 1 + (0.887 + 1.11i)T + (-19.8 + 86.7i)T^{2} \)
97 \( 1 + (-15.3 - 3.50i)T + (87.3 + 42.0i)T^{2} \)
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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

−10.32545204553035475734528134122, −9.118845265473354436827015493399, −8.279359475103562721561613357162, −7.982013434919503677384120139821, −7.32123906434506163200442655668, −5.86811765065795375865657910100, −5.27694949351462687089957931987, −3.79255999228030592810454587365, −2.37337820652177688046594700560, −0.59393026224928476511153861174, 1.53661490412266991596074230642, 2.29775728184295354990640619926, 3.71179518710871527337976682286, 5.07713762516442669502469638425, 5.89185367919709432029633538151, 7.43406176954442040315573216944, 8.274765234008113014538739715558, 8.808233410630483005972945155053, 9.500004352246892285236914178959, 10.57864943494223110313495004442

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