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Properties

Label 2-925-37.10-c1-0-20
Degree $2$
Conductor $925$
Sign $0.890 - 0.455i$
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  + (−0.704 + 1.22i)2-s + (−0.0771 − 0.133i)3-s + (0.00725 + 0.0125i)4-s + 0.217·6-s + (−1.20 − 2.08i)7-s − 2.83·8-s + (1.48 − 2.57i)9-s − 4.38·11-s + (0.00112 − 0.00194i)12-s + (1.99 + 3.45i)13-s + 3.39·14-s + (1.98 − 3.43i)16-s + (0.776 − 1.34i)17-s + (2.09 + 3.63i)18-s + (0.736 + 1.27i)19-s + ⋯
L(s)  = 1  + (−0.498 + 0.862i)2-s + (−0.0445 − 0.0771i)3-s + (0.00362 + 0.00628i)4-s + 0.0887·6-s + (−0.455 − 0.789i)7-s − 1.00·8-s + (0.496 − 0.859i)9-s − 1.32·11-s + (0.000323 − 0.000560i)12-s + (0.553 + 0.958i)13-s + 0.908·14-s + (0.496 − 0.859i)16-s + (0.188 − 0.326i)17-s + (0.494 + 0.856i)18-s + (0.168 + 0.292i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.890 - 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.890 - 0.455i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (676, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.890 - 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.05648 + 0.254437i\)
\(L(\frac12)\) \(\approx\) \(1.05648 + 0.254437i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.76 + 1.94i)T \)
good2 \( 1 + (0.704 - 1.22i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.0771 + 0.133i)T + (-1.5 + 2.59i)T^{2} \)
7 \( 1 + (1.20 + 2.08i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 4.38T + 11T^{2} \)
13 \( 1 + (-1.99 - 3.45i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.776 + 1.34i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.736 - 1.27i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 9.41T + 23T^{2} \)
29 \( 1 - 3.09T + 29T^{2} \)
31 \( 1 - 8.30T + 31T^{2} \)
41 \( 1 + (4.05 + 7.01i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 2.66T + 43T^{2} \)
47 \( 1 + 1.40T + 47T^{2} \)
53 \( 1 + (-4.36 + 7.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.94 - 6.82i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.760 + 1.31i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.41 + 4.18i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (4.93 + 8.54i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 + (-8.24 - 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.26 - 5.65i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-6.38 + 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.4T + 97T^{2} \)
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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

−9.920824509631618796326536372824, −9.189762358141939507933713720113, −8.373094562846591409982216277130, −7.42657457573440554441277555976, −6.88039960918934253768674949608, −6.21968170054397368639756599223, −5.01451199152353503125489890344, −3.77738848691483966562543248662, −2.79955390051627797099936921782, −0.76559932295392952075518213038, 1.06864257162427143586615858877, 2.62565249947027894721696545984, 2.99938442023484634681650020275, 4.80994068622798604514909336464, 5.58212899219694219887350276400, 6.49432106205305731399523624197, 7.77642141105617477160480097775, 8.474284665808948273298751548137, 9.392569425114095968878248013511, 10.22727708971603331367603687884

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