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Properties

Label 2-925-185.184-c1-0-44
Degree $2$
Conductor $925$
Sign $-0.588 - 0.808i$
Analytic cond. $7.38616$
Root an. cond. $2.71774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s − i·3-s + 2·4-s + 2i·6-s − 3i·7-s + 2·9-s − 3·11-s − 2i·12-s − 6·13-s + 6i·14-s − 4·16-s − 2·17-s − 4·18-s + 6i·19-s − 3·21-s + 6·22-s + ⋯
L(s)  = 1  − 1.41·2-s − 0.577i·3-s + 4-s + 0.816i·6-s − 1.13i·7-s + 0.666·9-s − 0.904·11-s − 0.577i·12-s − 1.66·13-s + 1.60i·14-s − 16-s − 0.485·17-s − 0.942·18-s + 1.37i·19-s − 0.654·21-s + 1.27·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.588 - 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.588 - 0.808i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (924, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.588 - 0.808i)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + (6 - i)T \)
good2 \( 1 + 2T + 2T^{2} \)
3 \( 1 + iT - 3T^{2} \)
7 \( 1 + 3iT - 7T^{2} \)
11 \( 1 + 3T + 11T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + 2T + 17T^{2} \)
19 \( 1 - 6iT - 19T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + 4iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
41 \( 1 + 3T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 + 4iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 12iT - 67T^{2} \)
71 \( 1 + 3T + 71T^{2} \)
73 \( 1 - 9iT - 73T^{2} \)
79 \( 1 - 6iT - 79T^{2} \)
83 \( 1 - 9iT - 83T^{2} \)
89 \( 1 + 14iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.856708578491769862192096362781, −8.568120197834974943947965229541, −7.73524956642737116298519328316, −7.31859599984709368908532779146, −6.69189628746302809585322722479, −5.11367360683593000206506120623, −4.12158679576840741710254206017, −2.44157127950063530936046869226, −1.33337464638965311817477155235, 0, 1.97141603022799877670491900210, 2.91404114078121048971487154030, 4.78708295854061940987114926688, 5.11731312134572542181451515571, 6.82019304187295289433647774834, 7.34602669372053426517570052138, 8.386104903181307551950930924229, 9.140132718435261482347671771552, 9.583629552497849715776819353613

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