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Properties

Label 2-925-185.174-c1-0-16
Degree $2$
Conductor $925$
Sign $0.610 + 0.792i$
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  + (−1.70 + 0.985i)2-s + (−2.29 − 1.32i)3-s + (0.943 − 1.63i)4-s + 5.22·6-s + (−0.678 − 0.391i)7-s − 0.221i·8-s + (2.00 + 3.48i)9-s − 0.568·11-s + (−4.33 + 2.50i)12-s + (−0.00529 − 0.00305i)13-s + 1.54·14-s + (2.10 + 3.64i)16-s + (2.56 − 1.48i)17-s + (−6.86 − 3.96i)18-s + (0.517 − 0.896i)19-s + ⋯
L(s)  = 1  + (−1.20 + 0.697i)2-s + (−1.32 − 0.764i)3-s + (0.471 − 0.817i)4-s + 2.13·6-s + (−0.256 − 0.148i)7-s − 0.0781i·8-s + (0.669 + 1.16i)9-s − 0.171·11-s + (−1.25 + 0.721i)12-s + (−0.00146 − 0.000847i)13-s + 0.412·14-s + (0.526 + 0.911i)16-s + (0.622 − 0.359i)17-s + (−1.61 − 0.933i)18-s + (0.118 − 0.205i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.792i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.610 + 0.792i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ 0.610 + 0.792i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305275 - 0.150150i\)
\(L(\frac12)\) \(\approx\) \(0.305275 - 0.150150i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.81 - 5.80i)T \)
good2 \( 1 + (1.70 - 0.985i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (2.29 + 1.32i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (0.678 + 0.391i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + 0.568T + 11T^{2} \)
13 \( 1 + (0.00529 + 0.00305i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.56 + 1.48i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.517 + 0.896i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 8.65iT - 23T^{2} \)
29 \( 1 - 4.06T + 29T^{2} \)
31 \( 1 + 9.38T + 31T^{2} \)
41 \( 1 + (-2.06 + 3.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 7.06iT - 43T^{2} \)
47 \( 1 - 5.52iT - 47T^{2} \)
53 \( 1 + (-7.91 + 4.57i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.62 - 8.00i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.99 + 8.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.83 - 5.67i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.96 - 3.40i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 16.5iT - 73T^{2} \)
79 \( 1 + (-5.45 + 9.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.745 - 0.430i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.143 + 0.248i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.33iT - 97T^{2} \)
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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

−9.895094757134469513989122834271, −9.109780043373429326893547145011, −8.055762856300765773515565975133, −7.22667770133769676673688654862, −6.88681038075725544405481115810, −5.82787361616320612833007806952, −5.24307708257620853136244517618, −3.58217002989676809002850114409, −1.58722269197475219670484379837, −0.42740588793098837177048658789, 0.852127691355025140594725367534, 2.45858033903261123625863470938, 3.86652837448457951627571155368, 5.03627464619762636378627668186, 5.77425164845581863742639321705, 6.76932546856699932515055910021, 7.958096504439650185998080801979, 8.810875852774290009364769169071, 9.691733185614502931106438412928, 10.24193342601547371743392311686

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