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Properties

Label 2-725-145.9-c1-0-24
Degree $2$
Conductor $725$
Sign $0.993 - 0.109i$
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  + (−2.22 + 1.06i)2-s + (0.731 + 0.917i)3-s + (2.54 − 3.18i)4-s + (−2.60 − 1.25i)6-s + (0.412 − 0.329i)7-s + (−1.13 + 4.98i)8-s + (0.361 − 1.58i)9-s + (2.26 − 0.516i)11-s + 4.78·12-s + (−3.52 + 0.803i)13-s + (−0.564 + 1.17i)14-s + (−0.992 − 4.34i)16-s + 3.45·17-s + (0.890 + 3.90i)18-s + (−5.57 − 4.44i)19-s + ⋯
L(s)  = 1  + (−1.57 + 0.756i)2-s + (0.422 + 0.529i)3-s + (1.27 − 1.59i)4-s + (−1.06 − 0.512i)6-s + (0.156 − 0.124i)7-s + (−0.402 + 1.76i)8-s + (0.120 − 0.527i)9-s + (0.682 − 0.155i)11-s + 1.38·12-s + (−0.976 + 0.222i)13-s + (−0.150 + 0.313i)14-s + (−0.248 − 1.08i)16-s + 0.838·17-s + (0.209 + 0.919i)18-s + (−1.27 − 1.01i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.109i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(725\)    =    \(5^{2} \cdot 29\)
Sign: $0.993 - 0.109i$
Analytic conductor: \(5.78915\)
Root analytic conductor: \(2.40606\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{725} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 725,\ (\ :1/2),\ 0.993 - 0.109i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.775129 + 0.0425768i\)
\(L(\frac12)\) \(\approx\) \(0.775129 + 0.0425768i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.34 + 0.625i)T \)
good2 \( 1 + (2.22 - 1.06i)T + (1.24 - 1.56i)T^{2} \)
3 \( 1 + (-0.731 - 0.917i)T + (-0.667 + 2.92i)T^{2} \)
7 \( 1 + (-0.412 + 0.329i)T + (1.55 - 6.82i)T^{2} \)
11 \( 1 + (-2.26 + 0.516i)T + (9.91 - 4.77i)T^{2} \)
13 \( 1 + (3.52 - 0.803i)T + (11.7 - 5.64i)T^{2} \)
17 \( 1 - 3.45T + 17T^{2} \)
19 \( 1 + (5.57 + 4.44i)T + (4.22 + 18.5i)T^{2} \)
23 \( 1 + (-1.75 + 3.64i)T + (-14.3 - 17.9i)T^{2} \)
31 \( 1 + (-1.35 - 2.80i)T + (-19.3 + 24.2i)T^{2} \)
37 \( 1 + (-0.971 + 4.25i)T + (-33.3 - 16.0i)T^{2} \)
41 \( 1 + 10.2iT - 41T^{2} \)
43 \( 1 + (6.11 + 2.94i)T + (26.8 + 33.6i)T^{2} \)
47 \( 1 + (1.50 + 6.57i)T + (-42.3 + 20.3i)T^{2} \)
53 \( 1 + (0.338 + 0.702i)T + (-33.0 + 41.4i)T^{2} \)
59 \( 1 - 6.23T + 59T^{2} \)
61 \( 1 + (-8.24 + 6.57i)T + (13.5 - 59.4i)T^{2} \)
67 \( 1 + (-3.52 - 0.803i)T + (60.3 + 29.0i)T^{2} \)
71 \( 1 + (-3.02 - 13.2i)T + (-63.9 + 30.8i)T^{2} \)
73 \( 1 + (4.33 + 2.08i)T + (45.5 + 57.0i)T^{2} \)
79 \( 1 + (-13.9 - 3.18i)T + (71.1 + 34.2i)T^{2} \)
83 \( 1 + (-7.26 - 5.79i)T + (18.4 + 80.9i)T^{2} \)
89 \( 1 + (-0.958 - 1.99i)T + (-55.4 + 69.5i)T^{2} \)
97 \( 1 + (5.16 - 6.47i)T + (-21.5 - 94.5i)T^{2} \)
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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

−10.13270397546675650049526838875, −9.356549518734012155653005736421, −8.777264960270279916535538817251, −8.113290211446708802135212972201, −6.86870080870497546946436283405, −6.62151420780355051774105225845, −5.15747802160911423909633978726, −3.93360801908201497250795204721, −2.36672600575405002803029885884, −0.71546311371327201747897318743, 1.29609904163550327443389945053, 2.20899844113643810102670759796, 3.25283575054277939112911040706, 4.81096564089707685722444063991, 6.40561182042105154463514607970, 7.38692913590290931487752898736, 8.071846842783664632893301668696, 8.552797961106113882760127679589, 9.757955688174974439479647699282, 10.05487715719039294984384218048

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