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Properties

Label 2-925-185.17-c1-0-40
Degree $2$
Conductor $925$
Sign $0.671 - 0.740i$
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  + (0.469 + 2.66i)2-s + (1.96 − 1.37i)3-s + (−4.99 + 1.81i)4-s + (4.57 + 4.57i)6-s + (−0.315 − 3.60i)7-s + (−4.48 − 7.77i)8-s + (0.933 − 2.56i)9-s + (1.47 − 0.851i)11-s + (−7.30 + 10.4i)12-s + (3.18 − 1.15i)13-s + (9.46 − 2.53i)14-s + (10.4 − 8.76i)16-s + (1.45 − 3.99i)17-s + (7.27 + 1.28i)18-s + (1.97 + 2.82i)19-s + ⋯
L(s)  = 1  + (0.332 + 1.88i)2-s + (1.13 − 0.792i)3-s + (−2.49 + 0.909i)4-s + (1.86 + 1.86i)6-s + (−0.119 − 1.36i)7-s + (−1.58 − 2.74i)8-s + (0.311 − 0.855i)9-s + (0.444 − 0.256i)11-s + (−2.10 + 3.00i)12-s + (0.882 − 0.321i)13-s + (2.52 − 0.677i)14-s + (2.61 − 2.19i)16-s + (0.352 − 0.969i)17-s + (1.71 + 0.302i)18-s + (0.453 + 0.647i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.671 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.11620 + 0.938062i\)
\(L(\frac12)\) \(\approx\) \(2.11620 + 0.938062i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.05 + 4.53i)T \)
good2 \( 1 + (-0.469 - 2.66i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-1.96 + 1.37i)T + (1.02 - 2.81i)T^{2} \)
7 \( 1 + (0.315 + 3.60i)T + (-6.89 + 1.21i)T^{2} \)
11 \( 1 + (-1.47 + 0.851i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-3.18 + 1.15i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.45 + 3.99i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (-1.97 - 2.82i)T + (-6.49 + 17.8i)T^{2} \)
23 \( 1 + (-1.74 + 3.02i)T + (-11.5 - 19.9i)T^{2} \)
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 \( 1 + 4.73T + 43T^{2} \)
47 \( 1 + (-2.31 + 0.620i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.491 - 5.61i)T + (-52.1 - 9.20i)T^{2} \)
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 \( 1 + (5.27 - 0.461i)T + (65.9 - 11.6i)T^{2} \)
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 \( 1 + (-4.25 + 1.98i)T + (53.3 - 63.5i)T^{2} \)
89 \( 1 + (13.9 + 1.21i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (6.80 + 3.92i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.663169211544370121450655778182, −8.841925199961201320966546434908, −8.250827007716141123099121464191, −7.33776491204833454465741149906, −7.16700299077646110077785940405, −6.18904910559436395067839263867, −5.12027469251458895793928753452, −3.90490501655053534126315452430, −3.29483968918637548721719256356, −0.988082575772485569750210788037, 1.63228358299449448265919437303, 2.59887338026903911269435834814, 3.41737965239376347184169987246, 4.07394386957432652146690367473, 5.11579434297815953314355825560, 6.09635044161589528443572023408, 8.141695088220542717091398253357, 8.802578509861642982099023829736, 9.403738676378766622211078389097, 9.779068781054348073205366584144

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