Properties

Label 2-925-185.17-c1-0-35
Degree $2$
Conductor $925$
Sign $0.293 - 0.955i$
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  + (0.321 + 1.82i)2-s + (2.27 − 1.59i)3-s + (−1.33 + 0.485i)4-s + (3.63 + 3.63i)6-s + (0.309 + 3.54i)7-s + (0.535 + 0.927i)8-s + (1.61 − 4.44i)9-s + (4.83 − 2.79i)11-s + (−2.26 + 3.23i)12-s + (−2.98 + 1.08i)13-s + (−6.35 + 1.70i)14-s + (−3.69 + 3.10i)16-s + (1.03 − 2.83i)17-s + (8.62 + 1.52i)18-s + (0.0922 + 0.131i)19-s + ⋯
L(s)  = 1  + (0.227 + 1.28i)2-s + (1.31 − 0.920i)3-s + (−0.667 + 0.242i)4-s + (1.48 + 1.48i)6-s + (0.117 + 1.33i)7-s + (0.189 + 0.327i)8-s + (0.539 − 1.48i)9-s + (1.45 − 0.841i)11-s + (−0.654 + 0.934i)12-s + (−0.826 + 0.300i)13-s + (−1.69 + 0.454i)14-s + (−0.923 + 0.775i)16-s + (0.250 − 0.687i)17-s + (2.03 + 0.358i)18-s + (0.0211 + 0.0302i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.293 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.293 - 0.955i$
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: \((2,\ 925,\ (\ :1/2),\ 0.293 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.42484 + 1.79177i\)
\(L(\frac12)\) \(\approx\) \(2.42484 + 1.79177i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 + 5.82i)T \)
good2 \( 1 + (-0.321 - 1.82i)T + (-1.87 + 0.684i)T^{2} \)
3 \( 1 + (-2.27 + 1.59i)T + (1.02 - 2.81i)T^{2} \)
7 \( 1 + (-0.309 - 3.54i)T + (-6.89 + 1.21i)T^{2} \)
11 \( 1 + (-4.83 + 2.79i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.98 - 1.08i)T + (9.95 - 8.35i)T^{2} \)
17 \( 1 + (-1.03 + 2.83i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (-0.0922 - 0.131i)T + (-6.49 + 17.8i)T^{2} \)
23 \( 1 + (3.22 - 5.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.02 + 7.54i)T + (-25.1 - 14.5i)T^{2} \)
31 \( 1 + (3.60 - 3.60i)T - 31iT^{2} \)
41 \( 1 + (-0.214 - 0.589i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 - 0.198T + 43T^{2} \)
47 \( 1 + (-0.516 + 0.138i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.212 + 2.42i)T + (-52.1 - 9.20i)T^{2} \)
59 \( 1 + (0.179 - 2.05i)T + (-58.1 - 10.2i)T^{2} \)
61 \( 1 + (0.187 - 0.402i)T + (-39.2 - 46.7i)T^{2} \)
67 \( 1 + (-14.0 + 1.22i)T + (65.9 - 11.6i)T^{2} \)
71 \( 1 + (-0.219 + 1.24i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (6.87 + 6.87i)T + 73iT^{2} \)
79 \( 1 + (6.64 - 0.581i)T + (77.7 - 13.7i)T^{2} \)
83 \( 1 + (8.81 - 4.10i)T + (53.3 - 63.5i)T^{2} \)
89 \( 1 + (5.12 + 0.448i)T + (87.6 + 15.4i)T^{2} \)
97 \( 1 + (6.38 + 3.68i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.570438433120014382676803572648, −9.045767222077290372314182430869, −8.394795758720229258858214402252, −7.67024036659984316150450204802, −6.94956613785394023065914656425, −6.13118725944491447498417545630, −5.37689301882945188155105838861, −3.95614661885224646747459835597, −2.70361686571543391646914128842, −1.75940930633555342653149518727, 1.38966285501135517765811641978, 2.51164556055765192931866269267, 3.59847183997583098846179751084, 4.11919472087178729052409131765, 4.71288005477762439543027161842, 6.73537628611169447756599294305, 7.49739646852980062059059385325, 8.537564214034163943563891180319, 9.461456591864001353395584236867, 10.06832309858248068245613222081

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