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Properties

Label 2-925-185.184-c1-0-40
Degree $2$
Conductor $925$
Sign $0.214 - 0.976i$
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  + 2.75·2-s + 2.91i·3-s + 5.58·4-s + 8.04i·6-s − 1.45i·7-s + 9.89·8-s − 5.51·9-s − 4.49·11-s + 16.3i·12-s + 3.36·13-s − 3.99i·14-s + 16.0·16-s + 0.0254·17-s − 15.2·18-s + 2.32i·19-s + ⋯
L(s)  = 1  + 1.94·2-s + 1.68i·3-s + 2.79·4-s + 3.28i·6-s − 0.548i·7-s + 3.49·8-s − 1.83·9-s − 1.35·11-s + 4.71i·12-s + 0.933·13-s − 1.06i·14-s + 4.01·16-s + 0.00617·17-s − 3.58·18-s + 0.534i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.214 - 0.976i$
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: \((2,\ 925,\ (\ :1/2),\ 0.214 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.94371 + 3.17080i\)
\(L(\frac12)\) \(\approx\) \(3.94371 + 3.17080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.72 + 3.82i)T \)
good2 \( 1 - 2.75T + 2T^{2} \)
3 \( 1 - 2.91iT - 3T^{2} \)
7 \( 1 + 1.45iT - 7T^{2} \)
11 \( 1 + 4.49T + 11T^{2} \)
13 \( 1 - 3.36T + 13T^{2} \)
17 \( 1 - 0.0254T + 17T^{2} \)
19 \( 1 - 2.32iT - 19T^{2} \)
23 \( 1 + 1.90T + 23T^{2} \)
29 \( 1 + 3.20iT - 29T^{2} \)
31 \( 1 + 2.69iT - 31T^{2} \)
41 \( 1 + 0.472T + 41T^{2} \)
43 \( 1 + 5.60T + 43T^{2} \)
47 \( 1 + 7.56iT - 47T^{2} \)
53 \( 1 + 7.83iT - 53T^{2} \)
59 \( 1 + 8.97iT - 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 - 3.88iT - 67T^{2} \)
71 \( 1 - 3.31T + 71T^{2} \)
73 \( 1 + 1.34iT - 73T^{2} \)
79 \( 1 - 16.9iT - 79T^{2} \)
83 \( 1 + 14.1iT - 83T^{2} \)
89 \( 1 - 10.7iT - 89T^{2} \)
97 \( 1 - 7.87T + 97T^{2} \)
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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

−10.38739018768267883833564545129, −10.06126698968060747529096272892, −8.473593871269555787193045211495, −7.52838610013291242334375821095, −6.30637322168427118203977309958, −5.45312305626487711128254490101, −4.93030519157914591141023851965, −3.88703613352321112805150890634, −3.56196075575078400297233215068, −2.34427905182124606398691925207, 1.55872589584247933462822254862, 2.53838143880884292010700161395, 3.26980347278748365035294506775, 4.79855415337074264821620171292, 5.65384304002637572143678876027, 6.24483960598163612313818877810, 7.05447171862893714581872373103, 7.75847257239009577493930489296, 8.584224347930558571549522101293, 10.47053014542542767866410157037

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