Properties

Label 2-925-37.28-c1-0-23
Degree $2$
Conductor $925$
Sign $-0.0342 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.17 + 0.384i)2-s + (−0.156 + 0.887i)3-s + (2.72 − 0.990i)4-s − 1.99i·6-s + (2.30 + 1.93i)7-s + (−1.71 + 0.991i)8-s + (2.05 + 0.748i)9-s + (2.41 + 4.18i)11-s + (0.453 + 2.56i)12-s + (1.66 + 4.58i)13-s + (−5.77 − 3.33i)14-s + (−1.07 + 0.902i)16-s + (2.75 − 7.56i)17-s + (−4.76 − 0.840i)18-s + (−2.22 − 0.392i)19-s + ⋯
L(s)  = 1  + (−1.54 + 0.271i)2-s + (−0.0903 + 0.512i)3-s + (1.36 − 0.495i)4-s − 0.813i·6-s + (0.872 + 0.732i)7-s + (−0.607 + 0.350i)8-s + (0.685 + 0.249i)9-s + (0.728 + 1.26i)11-s + (0.130 + 0.741i)12-s + (0.462 + 1.27i)13-s + (−1.54 − 0.891i)14-s + (−0.268 + 0.225i)16-s + (0.668 − 1.83i)17-s + (−1.12 − 0.198i)18-s + (−0.510 − 0.0899i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0342 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0342 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.0342 - 0.999i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.0342 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656519 + 0.679378i\)
\(L(\frac12)\) \(\approx\) \(0.656519 + 0.679378i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.34 + 5.61i)T \)
good2 \( 1 + (2.17 - 0.384i)T + (1.87 - 0.684i)T^{2} \)
3 \( 1 + (0.156 - 0.887i)T + (-2.81 - 1.02i)T^{2} \)
7 \( 1 + (-2.30 - 1.93i)T + (1.21 + 6.89i)T^{2} \)
11 \( 1 + (-2.41 - 4.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.66 - 4.58i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (-2.75 + 7.56i)T + (-13.0 - 10.9i)T^{2} \)
19 \( 1 + (2.22 + 0.392i)T + (17.8 + 6.49i)T^{2} \)
23 \( 1 + (-3.17 - 1.83i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.94 + 4.58i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 3.06iT - 31T^{2} \)
41 \( 1 + (-3.19 + 1.16i)T + (31.4 - 26.3i)T^{2} \)
43 \( 1 - 5.63iT - 43T^{2} \)
47 \( 1 + (1.75 - 3.03i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.43 + 2.04i)T + (9.20 - 52.1i)T^{2} \)
59 \( 1 + (6.41 + 7.64i)T + (-10.2 + 58.1i)T^{2} \)
61 \( 1 + (-1.00 - 2.77i)T + (-46.7 + 39.2i)T^{2} \)
67 \( 1 + (6.15 + 5.16i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (-1.74 + 9.90i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + (3.57 - 4.26i)T + (-13.7 - 77.7i)T^{2} \)
83 \( 1 + (-0.180 - 0.0655i)T + (63.5 + 53.3i)T^{2} \)
89 \( 1 + (0.630 + 0.751i)T + (-15.4 + 87.6i)T^{2} \)
97 \( 1 + (-4.09 - 2.36i)T + (48.5 + 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.854331225901578136225202116098, −9.439052619301269960120603782268, −8.903371556413025440800976472097, −7.80134062453898943805355413168, −7.20292583314381649101963705372, −6.36641798756352941895537301484, −4.87572616458367990494308736816, −4.31140300060978075472594330877, −2.29809906770173296356331069511, −1.33914881121258057035668707043, 1.02173727317524583977043998251, 1.37662390505392465299992147581, 3.15544275906259880626718227716, 4.35103440957957375661513975554, 5.87939397176836466193195309389, 6.75040208192578230568042928390, 7.63563617155290637578364531034, 8.483566965212223373493397012975, 8.588510796348434938092308325001, 10.12471434940541829938906749840

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