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Properties

Label 2-725-145.54-c1-0-17
Degree $2$
Conductor $725$
Sign $0.827 + 0.561i$
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  + (−1.21 − 0.277i)2-s + (0.781 + 1.62i)3-s + (−0.400 − 0.193i)4-s + (−0.5 − 2.19i)6-s + (−0.300 − 0.623i)7-s + (2.38 + 1.90i)8-s + (−0.153 + 0.193i)9-s + (−1.77 − 2.22i)11-s − 0.801i·12-s + (1.14 − 0.914i)13-s + (0.192 + 0.841i)14-s + (−1.81 − 2.27i)16-s − 1.60i·17-s + (0.240 − 0.192i)18-s + (−2.42 − 1.16i)19-s + ⋯
L(s)  = 1  + (−0.859 − 0.196i)2-s + (0.451 + 0.937i)3-s + (−0.200 − 0.0965i)4-s + (−0.204 − 0.894i)6-s + (−0.113 − 0.235i)7-s + (0.842 + 0.672i)8-s + (−0.0513 + 0.0643i)9-s + (−0.535 − 0.672i)11-s − 0.231i·12-s + (0.318 − 0.253i)13-s + (0.0513 + 0.224i)14-s + (−0.453 − 0.569i)16-s − 0.388i·17-s + (0.0567 − 0.0452i)18-s + (−0.556 − 0.267i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.885403 - 0.272133i\)
\(L(\frac12)\) \(\approx\) \(0.885403 - 0.272133i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.71 + 3.89i)T \)
good2 \( 1 + (1.21 + 0.277i)T + (1.80 + 0.867i)T^{2} \)
3 \( 1 + (-0.781 - 1.62i)T + (-1.87 + 2.34i)T^{2} \)
7 \( 1 + (0.300 + 0.623i)T + (-4.36 + 5.47i)T^{2} \)
11 \( 1 + (1.77 + 2.22i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-1.14 + 0.914i)T + (2.89 - 12.6i)T^{2} \)
17 \( 1 + 1.60iT - 17T^{2} \)
19 \( 1 + (2.42 + 1.16i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-5.02 + 1.14i)T + (20.7 - 9.97i)T^{2} \)
31 \( 1 + (-0.434 + 1.90i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (-2.22 - 1.77i)T + (8.23 + 36.0i)T^{2} \)
41 \( 1 - 6.49T + 41T^{2} \)
43 \( 1 + (0.648 - 0.147i)T + (38.7 - 18.6i)T^{2} \)
47 \( 1 + (-3.71 + 2.96i)T + (10.4 - 45.8i)T^{2} \)
53 \( 1 + (-0.0476 - 0.0108i)T + (47.7 + 22.9i)T^{2} \)
59 \( 1 - 6.39T + 59T^{2} \)
61 \( 1 + (-1.17 + 0.567i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-11.6 - 9.32i)T + (14.9 + 65.3i)T^{2} \)
71 \( 1 + (1.40 + 1.76i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (8.11 - 1.85i)T + (65.7 - 31.6i)T^{2} \)
79 \( 1 + (-6.07 + 7.61i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (-1.74 + 3.62i)T + (-51.7 - 64.8i)T^{2} \)
89 \( 1 + (-2.50 + 10.9i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (1.98 - 4.11i)T + (-60.4 - 75.8i)T^{2} \)
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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

−10.20889091316089239811614531022, −9.445470427177749964503732182289, −8.810857851477387226185964210478, −8.134577993792395851466778113158, −7.08183311932985673040075798301, −5.70762202020766340153238928515, −4.70781936847297616254960580016, −3.79733346468450352698774143272, −2.56914338507349949829996305524, −0.71700945976289060022506502668, 1.25774665699899364000769370177, 2.43316760190735844865230745259, 3.93261557651590049494530555198, 5.09523753271867131135582022187, 6.49023811466512824190417764450, 7.31831971545766980776078696261, 7.86515998115949357303684179776, 8.713560582140002374593123207487, 9.341181005146852501312386853298, 10.34247213966984925819632614743

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