↑

Properties

Label 2-175-5.4-c9-0-59
Degree $2$
Conductor $175$
Sign $0.894 - 0.447i$
Analytic cond. $90.1312$
Root an. cond. $9.49374$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 21.5i·2-s + 221. i·3-s + 48.6·4-s − 4.77e3·6-s − 2.40e3i·7-s + 1.20e4i·8-s − 2.95e4·9-s + 7.26e3·11-s + 1.07e4i·12-s − 1.04e5i·13-s + 5.16e4·14-s − 2.34e5·16-s − 3.77e5i·17-s − 6.36e5i·18-s − 6.99e5·19-s + ⋯
L(s)  = 1  + 0.951i·2-s + 1.58i·3-s + 0.0949·4-s − 1.50·6-s − 0.377i·7-s + 1.04i·8-s − 1.50·9-s + 0.149·11-s + 0.150i·12-s − 1.01i·13-s + 0.359·14-s − 0.896·16-s − 1.09i·17-s − 1.43i·18-s − 1.23·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.894 - 0.447i$
Analytic conductor: \(90.1312\)
Root analytic conductor: \(9.49374\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :9/2),\ 0.894 - 0.447i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.194947515\)
\(L(\frac12)\) \(\approx\) \(1.194947515\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
7 \( 1 + 2.40e3iT \)
good2 \( 1 - 21.5iT - 512T^{2} \)
3 \( 1 - 221. iT - 1.96e4T^{2} \)
11 \( 1 - 7.26e3T + 2.35e9T^{2} \)
13 \( 1 + 1.04e5iT - 1.06e10T^{2} \)
17 \( 1 + 3.77e5iT - 1.18e11T^{2} \)
19 \( 1 + 6.99e5T + 3.22e11T^{2} \)
23 \( 1 + 1.41e5iT - 1.80e12T^{2} \)
29 \( 1 + 4.83e6T + 1.45e13T^{2} \)
31 \( 1 - 9.21e6T + 2.64e13T^{2} \)
37 \( 1 + 1.62e7iT - 1.29e14T^{2} \)
41 \( 1 + 2.26e6T + 3.27e14T^{2} \)
43 \( 1 + 5.43e6iT - 5.02e14T^{2} \)
47 \( 1 + 5.34e7iT - 1.11e15T^{2} \)
53 \( 1 + 3.47e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.69e8T + 8.66e15T^{2} \)
61 \( 1 + 4.86e7T + 1.16e16T^{2} \)
67 \( 1 - 3.24e7iT - 2.72e16T^{2} \)
71 \( 1 + 1.15e8T + 4.58e16T^{2} \)
73 \( 1 + 2.66e8iT - 5.88e16T^{2} \)
79 \( 1 + 2.75e8T + 1.19e17T^{2} \)
83 \( 1 - 4.29e8iT - 1.86e17T^{2} \)
89 \( 1 - 8.82e8T + 3.50e17T^{2} \)
97 \( 1 - 3.64e8iT - 7.60e17T^{2} \)
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

−10.83646236843318588423249761719, −10.15040221348447683497103782796, −9.030209936224051171821120809795, −8.089323435920299683153158780904, −6.91976286368872467727805715828, −5.69243453473160780069042319751, −4.88601878875896052933442462984, −3.77284838340087680491465330201, −2.48547233579694860020231368209, −0.25730020738672799287783489334, 1.17561101203044414313066934816, 1.85915987158136244078472361494, 2.72137658706633495207964985953, 4.18558608873405764702268325529, 6.18488570592863993410312792041, 6.66047882669856324450412906134, 7.86193043426187611202233845731, 8.891813685224896686648318807907, 10.19692230014648414255829481187, 11.37032588238700139468656813508

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