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Properties

Label 2-925-185.84-c1-0-47
Degree $2$
Conductor $925$
Sign $-0.803 + 0.594i$
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  + (−1.23 − 0.711i)2-s + (2.52 − 1.45i)3-s + (0.0113 + 0.0197i)4-s − 4.14·6-s + (4.52 − 2.61i)7-s + 2.81i·8-s + (2.75 − 4.76i)9-s − 2.54·11-s + (0.0574 + 0.0331i)12-s + (−1.78 + 1.03i)13-s − 7.42·14-s + (2.02 − 3.50i)16-s + (−6.55 − 3.78i)17-s + (−6.77 + 3.91i)18-s + (−1.09 − 1.89i)19-s + ⋯
L(s)  = 1  + (−0.870 − 0.502i)2-s + (1.45 − 0.841i)3-s + (0.00569 + 0.00985i)4-s − 1.69·6-s + (1.70 − 0.986i)7-s + 0.994i·8-s + (0.917 − 1.58i)9-s − 0.766·11-s + (0.0165 + 0.00958i)12-s + (−0.495 + 0.286i)13-s − 1.98·14-s + (0.505 − 0.875i)16-s + (−1.59 − 0.918i)17-s + (−1.59 + 0.922i)18-s + (−0.251 − 0.435i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.803 + 0.594i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (824, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.803 + 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.516294 - 1.56545i\)
\(L(\frac12)\) \(\approx\) \(0.516294 - 1.56545i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.08 + 0.160i)T \)
good2 \( 1 + (1.23 + 0.711i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-2.52 + 1.45i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (-4.52 + 2.61i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + 2.54T + 11T^{2} \)
13 \( 1 + (1.78 - 1.03i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (6.55 + 3.78i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.09 + 1.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.86iT - 23T^{2} \)
29 \( 1 - 1.70T + 29T^{2} \)
31 \( 1 - 8.03T + 31T^{2} \)
41 \( 1 + (4.51 + 7.81i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 0.751iT - 43T^{2} \)
47 \( 1 - 0.0613iT - 47T^{2} \)
53 \( 1 + (-1.68 - 0.973i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.44 + 4.24i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.63 - 2.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.42 - 5.43i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.10 - 1.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 9.02iT - 73T^{2} \)
79 \( 1 + (-3.42 - 5.93i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-7.41 - 4.27i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-0.806 + 1.39i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.26iT - 97T^{2} \)
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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

−9.580890291463877083835579217874, −8.728701639808838449351597418801, −8.234497621772971537753259926155, −7.55086843840852089860075727409, −6.92173231151683594812383185827, −5.08380335815490089462005541188, −4.30010308855489446363534714486, −2.59993555525540918570824572240, −2.01999041493942777782432533904, −0.896880031583287508548183256112, 1.98373415194952765651146404093, 2.85392131442841942242042457741, 4.34834153528637583678736842535, 4.77372883815902134287446345722, 6.29120060819819175905848595961, 7.72612366355219380023755272580, 8.170284837585344967638246847932, 8.572577755824317100655774014233, 9.195206024509289589778249400649, 10.18633821161926155988487482043

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