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Properties

Label 2-725-145.4-c1-0-39
Degree $2$
Conductor $725$
Sign $0.180 + 0.983i$
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  + (0.122 − 0.536i)2-s + (1.77 − 0.855i)3-s + (1.52 + 0.736i)4-s + (−0.241 − 1.05i)6-s + (−1.94 − 4.03i)7-s + (1.26 − 1.58i)8-s + (0.556 − 0.697i)9-s + (1.16 − 0.929i)11-s + 3.34·12-s + (1.13 − 0.906i)13-s + (−2.39 + 0.547i)14-s + (1.41 + 1.78i)16-s − 2.07·17-s + (−0.305 − 0.383i)18-s + (0.236 − 0.490i)19-s + ⋯
L(s)  = 1  + (0.0865 − 0.379i)2-s + (1.02 − 0.494i)3-s + (0.764 + 0.368i)4-s + (−0.0985 − 0.431i)6-s + (−0.733 − 1.52i)7-s + (0.448 − 0.562i)8-s + (0.185 − 0.232i)9-s + (0.351 − 0.280i)11-s + 0.966·12-s + (0.315 − 0.251i)13-s + (−0.640 + 0.146i)14-s + (0.354 + 0.445i)16-s − 0.502·17-s + (−0.0720 − 0.0903i)18-s + (0.0541 − 0.112i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.180 + 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.180 + 0.983i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (149, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.180 + 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92612 - 1.60439i\)
\(L(\frac12)\) \(\approx\) \(1.92612 - 1.60439i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.78 - 3.83i)T \)
good2 \( 1 + (-0.122 + 0.536i)T + (-1.80 - 0.867i)T^{2} \)
3 \( 1 + (-1.77 + 0.855i)T + (1.87 - 2.34i)T^{2} \)
7 \( 1 + (1.94 + 4.03i)T + (-4.36 + 5.47i)T^{2} \)
11 \( 1 + (-1.16 + 0.929i)T + (2.44 - 10.7i)T^{2} \)
13 \( 1 + (-1.13 + 0.906i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + 2.07T + 17T^{2} \)
19 \( 1 + (-0.236 + 0.490i)T + (-11.8 - 14.8i)T^{2} \)
23 \( 1 + (1.67 - 0.382i)T + (20.7 - 9.97i)T^{2} \)
31 \( 1 + (4.54 + 1.03i)T + (27.9 + 13.4i)T^{2} \)
37 \( 1 + (-1.89 + 2.37i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 + 0.595iT - 41T^{2} \)
43 \( 1 + (0.0733 + 0.321i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-7.15 - 8.97i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (-10.0 - 2.28i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 - 4.03T + 59T^{2} \)
61 \( 1 + (-5.18 - 10.7i)T + (-38.0 + 47.6i)T^{2} \)
67 \( 1 + (6.52 + 5.20i)T + (14.9 + 65.3i)T^{2} \)
71 \( 1 + (5.78 + 7.25i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (-0.769 - 3.37i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (6.50 + 5.19i)T + (17.5 + 77.0i)T^{2} \)
83 \( 1 + (0.460 - 0.955i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (4.68 + 1.06i)T + (80.1 + 38.6i)T^{2} \)
97 \( 1 + (-4.35 - 2.09i)T + (60.4 + 75.8i)T^{2} \)
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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

−10.45375048869224404525505504568, −9.321367774640382897363848273692, −8.401966447262272138285053367370, −7.40379715172446183614831961107, −7.09277903397776002071559598369, −6.04467821776745994061341879317, −4.16904077275672337962190935025, −3.45191274264293956170669834879, −2.56174619753331704846879750784, −1.21163137301577378012171831782, 2.10776576882758514605329242917, 2.81541937946615817285472348030, 3.99534878373735619115876030893, 5.40253922456964125026909902281, 6.18719869523818802036635189720, 6.96900618658734194122839670739, 8.240342546381645149334340416587, 8.844277363080204069989168110449, 9.599717017786097129849996157338, 10.32151980757035494514367792226

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