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Properties

Label 2-925-185.43-c1-0-30
Degree $2$
Conductor $925$
Sign $0.708 - 0.705i$
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  + 2.42i·2-s + (−1.67 + 1.67i)3-s − 3.90·4-s + (−4.06 − 4.06i)6-s + (−1.37 + 1.37i)7-s − 4.62i·8-s − 2.60i·9-s − 4.19i·11-s + (6.52 − 6.52i)12-s − 3.09i·13-s + (−3.34 − 3.34i)14-s + 3.42·16-s + 4.21·17-s + 6.31·18-s + (−2.00 + 2.00i)19-s + ⋯
L(s)  = 1  + 1.71i·2-s + (−0.966 + 0.966i)3-s − 1.95·4-s + (−1.65 − 1.65i)6-s + (−0.519 + 0.519i)7-s − 1.63i·8-s − 0.866i·9-s − 1.26i·11-s + (1.88 − 1.88i)12-s − 0.859i·13-s + (−0.893 − 0.893i)14-s + 0.855·16-s + 1.02·17-s + 1.48·18-s + (−0.459 + 0.459i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 - 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.708 - 0.705i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.708 - 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.324724 + 0.134134i\)
\(L(\frac12)\) \(\approx\) \(0.324724 + 0.134134i\)
\(L(\frac{3}{2})\) 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 \( 1 + (3.09 + 5.23i)T \)
good2 \( 1 - 2.42iT - 2T^{2} \)
3 \( 1 + (1.67 - 1.67i)T - 3iT^{2} \)
7 \( 1 + (1.37 - 1.37i)T - 7iT^{2} \)
11 \( 1 + 4.19iT - 11T^{2} \)
13 \( 1 + 3.09iT - 13T^{2} \)
17 \( 1 - 4.21T + 17T^{2} \)
19 \( 1 + (2.00 - 2.00i)T - 19iT^{2} \)
23 \( 1 - 2.97iT - 23T^{2} \)
29 \( 1 + (6.59 + 6.59i)T + 29iT^{2} \)
31 \( 1 + (-5.90 + 5.90i)T - 31iT^{2} \)
41 \( 1 - 3.84iT - 41T^{2} \)
43 \( 1 - 10.3iT - 43T^{2} \)
47 \( 1 + (3.77 - 3.77i)T - 47iT^{2} \)
53 \( 1 + (6.25 + 6.25i)T + 53iT^{2} \)
59 \( 1 + (-0.919 + 0.919i)T - 59iT^{2} \)
61 \( 1 + (-0.985 + 0.985i)T - 61iT^{2} \)
67 \( 1 + (9.13 + 9.13i)T + 67iT^{2} \)
71 \( 1 - 4.32T + 71T^{2} \)
73 \( 1 + (-2.68 + 2.68i)T - 73iT^{2} \)
79 \( 1 + (-2.71 + 2.71i)T - 79iT^{2} \)
83 \( 1 + (-0.768 - 0.768i)T + 83iT^{2} \)
89 \( 1 + (10.6 + 10.6i)T + 89iT^{2} \)
97 \( 1 + 19.6T + 97T^{2} \)
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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

−9.791736851494414168699672889789, −9.434251503095325987018553146705, −8.167791032335714836506532693612, −7.78606867579194239367608868361, −6.24323658794725418475485677109, −5.92618152584904902252907603037, −5.38395288825596392160941160160, −4.34496463079735813891824293507, −3.27860745827513167663372506481, −0.21529551177470635914654984633, 1.18543315997703843738452585883, 2.06225108962111176162191403438, 3.40461476648079697016895165643, 4.44937261490143313022371262539, 5.36284930142550460920032316267, 6.80572205938311409430338386231, 7.09132741779057161959292589100, 8.578923679348850849193935580471, 9.542977068431193890695053993334, 10.24975858710215834474909958968

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