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Properties

Label 2-925-185.84-c1-0-28
Degree $2$
Conductor $925$
Sign $-0.662 + 0.748i$
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  + (−0.268 − 0.155i)2-s + (−1.78 + 1.03i)3-s + (−0.951 − 1.64i)4-s + 0.639·6-s + (−1.83 + 1.06i)7-s + 1.21i·8-s + (0.625 − 1.08i)9-s + 5.48·11-s + (3.39 + 1.96i)12-s + (−1.43 + 0.826i)13-s + 0.657·14-s + (−1.71 + 2.97i)16-s + (2.46 + 1.42i)17-s + (−0.336 + 0.194i)18-s + (−1.19 − 2.07i)19-s + ⋯
L(s)  = 1  + (−0.189 − 0.109i)2-s + (−1.03 + 0.595i)3-s + (−0.475 − 0.824i)4-s + 0.261·6-s + (−0.694 + 0.400i)7-s + 0.428i·8-s + (0.208 − 0.361i)9-s + 1.65·11-s + (0.981 + 0.566i)12-s + (−0.396 + 0.229i)13-s + 0.175·14-s + (−0.428 + 0.742i)16-s + (0.597 + 0.344i)17-s + (−0.0792 + 0.0457i)18-s + (−0.274 − 0.475i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 + 0.748i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.662 + 0.748i$
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: \((2,\ 925,\ (\ :1/2),\ -0.662 + 0.748i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.102394 - 0.227381i\)
\(L(\frac12)\) \(\approx\) \(0.102394 - 0.227381i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.58 + 2.41i)T \)
good2 \( 1 + (0.268 + 0.155i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (1.78 - 1.03i)T + (1.5 - 2.59i)T^{2} \)
7 \( 1 + (1.83 - 1.06i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 - 5.48T + 11T^{2} \)
13 \( 1 + (1.43 - 0.826i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2.46 - 1.42i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.19 + 2.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 1.29iT - 23T^{2} \)
29 \( 1 - 0.459T + 29T^{2} \)
31 \( 1 + 7.00T + 31T^{2} \)
41 \( 1 + (3.63 + 6.30i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 4.39iT - 43T^{2} \)
47 \( 1 + 0.310iT - 47T^{2} \)
53 \( 1 + (10.4 + 6.00i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.14 + 7.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.10 + 8.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.15 + 0.664i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (0.293 + 0.508i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 10.8iT - 73T^{2} \)
79 \( 1 + (1.41 + 2.45i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.2 - 5.91i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (8.40 - 14.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 18.5iT - 97T^{2} \)
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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

−9.558258190548570367687054450526, −9.443028827678422913176260188024, −8.390154872011190832320053055054, −6.81340183438688430640526075694, −6.19998637914096091226708541919, −5.40883872987640202470046780853, −4.59559560512096937985890262240, −3.59733014041589315935024162416, −1.76970310919029807189969529236, −0.16364161351489132651224815950, 1.25245545169751075523018779074, 3.26225588940137241667827629489, 4.03327880389112697836869232412, 5.26238104581986999968062558169, 6.32143594989224630503666271511, 6.93908357083002224641247401002, 7.61502272580602579184364829681, 8.795714446025745370709035247853, 9.463334977770762686421391174339, 10.31473936438571452799261847957

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