Properties

Label 2-175-5.4-c9-0-56
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.17i·2-s + 65.7i·3-s + 427.·4-s − 602.·6-s − 2.40e3i·7-s + 8.62e3i·8-s + 1.53e4·9-s + 3.50e4·11-s + 2.81e4i·12-s − 7.74e4i·13-s + 2.20e4·14-s + 1.40e5·16-s + 2.29e5i·17-s + 1.40e5i·18-s − 1.64e4·19-s + ⋯
L(s)  = 1  + 0.405i·2-s + 0.468i·3-s + 0.835·4-s − 0.189·6-s − 0.377i·7-s + 0.744i·8-s + 0.780·9-s + 0.722·11-s + 0.391i·12-s − 0.751i·13-s + 0.153·14-s + 0.534·16-s + 0.667i·17-s + 0.316i·18-s − 0.0289·19-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

\(L(5)\) \(\approx\) \(3.393393871\)
\(L(\frac12)\) \(\approx\) \(3.393393871\)
\(L(\frac{11}{2})\) 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 \( 1 - 9.17iT - 512T^{2} \)
3 \( 1 - 65.7iT - 1.96e4T^{2} \)
11 \( 1 - 3.50e4T + 2.35e9T^{2} \)
13 \( 1 + 7.74e4iT - 1.06e10T^{2} \)
17 \( 1 - 2.29e5iT - 1.18e11T^{2} \)
19 \( 1 + 1.64e4T + 3.22e11T^{2} \)
23 \( 1 + 2.57e6iT - 1.80e12T^{2} \)
29 \( 1 - 6.62e6T + 1.45e13T^{2} \)
31 \( 1 + 8.17e6T + 2.64e13T^{2} \)
37 \( 1 + 9.70e6iT - 1.29e14T^{2} \)
41 \( 1 - 2.98e7T + 3.27e14T^{2} \)
43 \( 1 + 1.95e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.93e6iT - 1.11e15T^{2} \)
53 \( 1 + 2.74e7iT - 3.29e15T^{2} \)
59 \( 1 + 5.24e7T + 8.66e15T^{2} \)
61 \( 1 - 2.23e7T + 1.16e16T^{2} \)
67 \( 1 + 2.74e8iT - 2.72e16T^{2} \)
71 \( 1 + 3.63e8T + 4.58e16T^{2} \)
73 \( 1 - 2.09e7iT - 5.88e16T^{2} \)
79 \( 1 - 2.65e8T + 1.19e17T^{2} \)
83 \( 1 + 9.43e6iT - 1.86e17T^{2} \)
89 \( 1 - 6.64e8T + 3.50e17T^{2} \)
97 \( 1 - 1.20e9iT - 7.60e17T^{2} \)
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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

−10.70005571229893331206051336692, −10.47555157863818443061246950676, −9.044400040307146856468111990826, −7.896922861561510456526427806861, −6.91762203136181381388601313142, −6.05574910262246413674219192346, −4.69411050701293414128489150857, −3.57334522278522012608041365580, −2.16403573641047107355625806084, −0.850483780253857071118536812771, 1.09698891795236830510446499656, 1.81527816743184604972810723403, 3.05331539304796349699928701768, 4.33104031023771865912467563076, 5.93508639461919858415672738119, 6.91083334739397695704854408381, 7.60285418988175606957842483149, 9.156241890429381949129870760344, 9.967648509971858461827038349106, 11.24938084171892409958277580191

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