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Properties

Label 2-925-185.128-c1-0-47
Degree $2$
Conductor $925$
Sign $0.932 - 0.362i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
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  + (2.66 − 0.469i)2-s + (1.37 + 1.96i)3-s + (4.99 − 1.81i)4-s + (4.57 + 4.57i)6-s + (−3.60 + 0.315i)7-s + (7.77 − 4.48i)8-s + (−0.933 + 2.56i)9-s + (1.47 − 0.851i)11-s + (10.4 + 7.30i)12-s + (1.15 + 3.18i)13-s + (−9.46 + 2.53i)14-s + (10.4 − 8.76i)16-s + (−3.99 − 1.45i)17-s + (−1.28 + 7.27i)18-s + (−1.97 − 2.82i)19-s + ⋯
L(s)  = 1  + (1.88 − 0.332i)2-s + (0.792 + 1.13i)3-s + (2.49 − 0.909i)4-s + (1.86 + 1.86i)6-s + (−1.36 + 0.119i)7-s + (2.74 − 1.58i)8-s + (−0.311 + 0.855i)9-s + (0.444 − 0.256i)11-s + (3.00 + 2.10i)12-s + (0.321 + 0.882i)13-s + (−2.52 + 0.677i)14-s + (2.61 − 2.19i)16-s + (−0.969 − 0.352i)17-s + (−0.302 + 1.71i)18-s + (−0.453 − 0.647i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.932 - 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.932 - 0.362i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (868, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.932 - 0.362i)\)

Particular Values

\(L(1)\) \(\approx\) \(5.54554 + 1.04022i\)
\(L(\frac12)\) \(\approx\) \(5.54554 + 1.04022i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
37 \( 1 + (4.53 + 4.05i)T \)
good2 \( 1 + (-2.66 + 0.469i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (-1.37 - 1.96i)T + (-1.02 + 2.81i)T^{2} \)
7 \( 1 + (3.60 - 0.315i)T + (6.89 - 1.21i)T^{2} \)
11 \( 1 + (-1.47 + 0.851i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.15 - 3.18i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.99 + 1.45i)T + (13.0 + 10.9i)T^{2} \)
19 \( 1 + (1.97 + 2.82i)T + (-6.49 + 17.8i)T^{2} \)
23 \( 1 + (-3.02 - 1.74i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.442 + 1.65i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (5.32 - 5.32i)T - 31iT^{2} \)
41 \( 1 + (1.37 + 3.77i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + 4.73iT - 43T^{2} \)
47 \( 1 + (0.620 + 2.31i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (5.61 + 0.491i)T + (52.1 + 9.20i)T^{2} \)
59 \( 1 + (-0.132 + 1.51i)T + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (-2.41 + 5.18i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (-0.461 - 5.27i)T + (-65.9 + 11.6i)T^{2} \)
71 \( 1 + (0.493 - 2.80i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (5.50 - 5.50i)T - 73iT^{2} \)
79 \( 1 + (7.89 - 0.691i)T + (77.7 - 13.7i)T^{2} \)
83 \( 1 + (-1.98 - 4.25i)T + (-53.3 + 63.5i)T^{2} \)
89 \( 1 + (-13.9 - 1.21i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (3.92 - 6.80i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.36014934986491977349979713103, −9.348723536929771907885558000631, −8.853396323238287067533784149879, −6.94931572390289006598491356295, −6.60176937570623949609576931590, −5.45032911006958098942425041682, −4.50080160006496776361730169293, −3.74278129966262903677035960309, −3.20908608396428290990593416254, −2.19213985371280718313228687832, 1.81342701326408219571534996117, 2.91527221607121002557707813998, 3.50466147188946446746417763006, 4.59927943818709786419739178850, 5.96871889824636722388285651961, 6.46088805341170163176911952403, 7.12864329353667606971031636420, 7.913122305796837488146917086989, 8.897502092205277647667716566542, 10.25487864033792822712277202437

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