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Properties

Label 2-925-185.184-c1-0-15
Degree $2$
Conductor $925$
Sign $0.672 - 0.740i$
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  − 2.59·2-s − 0.695i·3-s + 4.75·4-s + 1.80i·6-s + 3.94i·7-s − 7.15·8-s + 2.51·9-s + 0.718·11-s − 3.30i·12-s + 6.10·13-s − 10.2i·14-s + 9.09·16-s − 2.79·17-s − 6.53·18-s − 0.449i·19-s + ⋯
L(s)  = 1  − 1.83·2-s − 0.401i·3-s + 2.37·4-s + 0.738i·6-s + 1.49i·7-s − 2.53·8-s + 0.838·9-s + 0.216·11-s − 0.954i·12-s + 1.69·13-s − 2.74i·14-s + 2.27·16-s − 0.678·17-s − 1.54·18-s − 0.103i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.672 - 0.740i$
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: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.672 - 0.740i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.679196 + 0.300652i\)
\(L(\frac12)\) \(\approx\) \(0.679196 + 0.300652i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.19 + 5.67i)T \)
good2 \( 1 + 2.59T + 2T^{2} \)
3 \( 1 + 0.695iT - 3T^{2} \)
7 \( 1 - 3.94iT - 7T^{2} \)
11 \( 1 - 0.718T + 11T^{2} \)
13 \( 1 - 6.10T + 13T^{2} \)
17 \( 1 + 2.79T + 17T^{2} \)
19 \( 1 + 0.449iT - 19T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 - 5.64iT - 29T^{2} \)
31 \( 1 - 9.09iT - 31T^{2} \)
41 \( 1 + 10.2T + 41T^{2} \)
43 \( 1 - 0.724T + 43T^{2} \)
47 \( 1 - 8.44iT - 47T^{2} \)
53 \( 1 + 8.18iT - 53T^{2} \)
59 \( 1 + 6.30iT - 59T^{2} \)
61 \( 1 + 5.93iT - 61T^{2} \)
67 \( 1 - 0.420iT - 67T^{2} \)
71 \( 1 - 12.2T + 71T^{2} \)
73 \( 1 - 13.1iT - 73T^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 + 3.21iT - 83T^{2} \)
89 \( 1 - 3.67iT - 89T^{2} \)
97 \( 1 - 1.64T + 97T^{2} \)
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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

−10.01330899819208157677564826865, −8.984637113526532980495902097625, −8.777019974900965370856966292703, −7.995428113235797463843666838397, −6.77396056246004283244155417401, −6.50459294932344070575581297754, −5.26211028488187400978647077871, −3.38310104255071644446371732494, −2.09415602463459257658665729901, −1.26673367994713160884393431810, 0.75075374196111079341424276834, 1.74775927096951201481880106768, 3.49982885202381585634663365151, 4.37865897891029657636090444229, 6.17286846673501727050598474572, 6.87744194384733438756731864480, 7.59249765928006288110597604546, 8.410086182394599708826353718181, 9.185561381994201546028785433508, 10.03191776511943453020712241067

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