↑

Properties

Label 2-925-185.23-c1-0-34
Degree $2$
Conductor $925$
Sign $-0.781 + 0.623i$
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  + (0.305 + 0.176i)2-s + (−2.52 − 0.676i)3-s + (−0.937 − 1.62i)4-s + (−0.652 − 0.652i)6-s + (1.50 + 0.404i)7-s − 1.36i·8-s + (3.31 + 1.91i)9-s − 2.57i·11-s + (1.26 + 4.73i)12-s + (4.41 − 2.54i)13-s + (0.389 + 0.389i)14-s + (−1.63 + 2.83i)16-s + (0.342 − 0.593i)17-s + (0.676 + 1.17i)18-s + (4.59 + 1.23i)19-s + ⋯
L(s)  = 1  + (0.216 + 0.124i)2-s + (−1.45 − 0.390i)3-s + (−0.468 − 0.812i)4-s + (−0.266 − 0.266i)6-s + (0.570 + 0.152i)7-s − 0.483i·8-s + (1.10 + 0.638i)9-s − 0.775i·11-s + (0.366 + 1.36i)12-s + (1.22 − 0.706i)13-s + (0.104 + 0.104i)14-s + (−0.408 + 0.707i)16-s + (0.0830 − 0.143i)17-s + (0.159 + 0.276i)18-s + (1.05 + 0.282i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 + 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(925\)    =    \(5^{2} \cdot 37\)
Sign: $-0.781 + 0.623i$
Analytic conductor: \(7.38616\)
Root analytic conductor: \(2.71774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{925} (393, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 925,\ (\ :1/2),\ -0.781 + 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.262066 - 0.748602i\)
\(L(\frac12)\) \(\approx\) \(0.262066 - 0.748602i\)
\(L(\frac{3}{2})\) 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 \( 1 + (4.74 - 3.80i)T \)
good2 \( 1 + (-0.305 - 0.176i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (2.52 + 0.676i)T + (2.59 + 1.5i)T^{2} \)
7 \( 1 + (-1.50 - 0.404i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + 2.57iT - 11T^{2} \)
13 \( 1 + (-4.41 + 2.54i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.342 + 0.593i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.59 - 1.23i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 5.83iT - 23T^{2} \)
29 \( 1 + (4.71 + 4.71i)T + 29iT^{2} \)
31 \( 1 + (1.40 - 1.40i)T - 31iT^{2} \)
41 \( 1 + (6.82 - 3.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + 2.28iT - 43T^{2} \)
47 \( 1 + (-7.99 + 7.99i)T - 47iT^{2} \)
53 \( 1 + (4.37 - 1.17i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.43 + 9.07i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.49 + 0.667i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.100 - 0.376i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-0.0695 - 0.120i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (6.70 - 6.70i)T - 73iT^{2} \)
79 \( 1 + (-0.455 - 0.121i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (-3.06 + 0.821i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + (-15.2 + 4.08i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + 13.7T + 97T^{2} \)
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

−10.06963502072256398876830703810, −8.866487381994117394605993204884, −8.064048173249940981574832664726, −6.80820546626769313818501478286, −6.04509684647292115319911953549, −5.49672984952491818312410134685, −4.84263444012563386262186157595, −3.55152304612546889850042156442, −1.48856480591521096347570564042, −0.48256584388106861749892128907, 1.51780286683293953238111956946, 3.45804912673155865850484413702, 4.30970286823963364723218108333, 5.09579919265844630315532520776, 5.81470719538216728163189076715, 7.02677774832309794091535343500, 7.69619078482517594829734419627, 8.913064888919257806329655766044, 9.562201520589889571164209371474, 10.75772482853968257654967274037

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