↑

Properties

Label 2-925-185.174-c1-0-12
Degree $2$
Conductor $925$
Sign $0.107 - 0.994i$
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  + (−2.18 + 1.26i)2-s + (−2.76 − 1.59i)3-s + (2.19 − 3.79i)4-s + 8.05·6-s + (2.85 + 1.64i)7-s + 6.02i·8-s + (3.58 + 6.20i)9-s + 3.95·11-s + (−12.1 + 6.99i)12-s + (4.56 + 2.63i)13-s − 8.31·14-s + (−3.23 − 5.59i)16-s + (2.02 − 1.17i)17-s + (−15.6 − 9.05i)18-s + (−2.43 + 4.21i)19-s + ⋯
L(s)  = 1  + (−1.54 + 0.893i)2-s + (−1.59 − 0.920i)3-s + (1.09 − 1.89i)4-s + 3.28·6-s + (1.07 + 0.621i)7-s + 2.13i·8-s + (1.19 + 2.06i)9-s + 1.19·11-s + (−3.49 + 2.01i)12-s + (1.26 + 0.730i)13-s − 2.22·14-s + (−0.807 − 1.39i)16-s + (0.491 − 0.283i)17-s + (−3.69 − 2.13i)18-s + (−0.558 + 0.966i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.107 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $0.107 - 0.994i$
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: \((2,\ 925,\ (\ :1/2),\ 0.107 - 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.374439 + 0.336285i\)
\(L(\frac12)\) \(\approx\) \(0.374439 + 0.336285i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.93 - 5.32i)T \)
good2 \( 1 + (2.18 - 1.26i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (2.76 + 1.59i)T + (1.5 + 2.59i)T^{2} \)
7 \( 1 + (-2.85 - 1.64i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 - 3.95T + 11T^{2} \)
13 \( 1 + (-4.56 - 2.63i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.02 + 1.17i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.43 - 4.21i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + 1.39iT - 23T^{2} \)
29 \( 1 + 3.38T + 29T^{2} \)
31 \( 1 - 5.02T + 31T^{2} \)
41 \( 1 + (5.32 - 9.22i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 - 7.67iT - 43T^{2} \)
47 \( 1 + 1.52iT - 47T^{2} \)
53 \( 1 + (2.89 - 1.67i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.74 + 3.01i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.659 + 1.14i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.0 + 6.40i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (3.64 - 6.30i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8.70iT - 73T^{2} \)
79 \( 1 + (-3.07 + 5.32i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.94 + 3.43i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.199 + 0.345i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.2iT - 97T^{2} \)
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

−10.26119424506930452135877189388, −9.289175196795465918619339334229, −8.274437997258886978859793030323, −7.88455899314782975044636157455, −6.66430747035251577611624500712, −6.35296853286717880864747267629, −5.64742120074391178421480453711, −4.54112901234809813946549665146, −1.63378790128305384985619170621, −1.26147061539084457075251122507, 0.62674859931536640300455292387, 1.50379407221916808486651944938, 3.57448439600223302511927007627, 4.30135047927572152512888735281, 5.54449059309803990165932321615, 6.56575275480833203732866289116, 7.50171423007617607929954180235, 8.597647663245507273281552728517, 9.235171032362380792559650401917, 10.18398153193426750464940420622

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