↑

Properties

Label 2-175-5.4-c3-0-10
Degree $2$
Conductor $175$
Sign $0.447 + 0.894i$
Analytic cond. $10.3253$
Root an. cond. $3.21330$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4.87i·2-s + 4.14i·3-s − 15.7·4-s + 20.2·6-s + 7i·7-s + 37.6i·8-s + 9.78·9-s + 36.9·11-s − 65.2i·12-s + 61.3i·13-s + 34.1·14-s + 57.7·16-s − 44.8i·17-s − 47.6i·18-s + 139.·19-s + ⋯
L(s)  = 1  − 1.72i·2-s + 0.798i·3-s − 1.96·4-s + 1.37·6-s + 0.377i·7-s + 1.66i·8-s + 0.362·9-s + 1.01·11-s − 1.57i·12-s + 1.30i·13-s + 0.651·14-s + 0.902·16-s − 0.639i·17-s − 0.624i·18-s + 1.68·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(175\)    =    \(5^{2} \cdot 7\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(10.3253\)
Root analytic conductor: \(3.21330\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{175} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 175,\ (\ :3/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.43295 - 0.885612i\)
\(L(\frac12)\) \(\approx\) \(1.43295 - 0.885612i\)
\(L(\frac{5}{2})\) 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 \( 1 - 7iT \)
good2 \( 1 + 4.87iT - 8T^{2} \)
3 \( 1 - 4.14iT - 27T^{2} \)
11 \( 1 - 36.9T + 1.33e3T^{2} \)
13 \( 1 - 61.3iT - 2.19e3T^{2} \)
17 \( 1 + 44.8iT - 4.91e3T^{2} \)
19 \( 1 - 139.T + 6.85e3T^{2} \)
23 \( 1 + 217. iT - 1.21e4T^{2} \)
29 \( 1 - 33.8T + 2.43e4T^{2} \)
31 \( 1 - 124.T + 2.97e4T^{2} \)
37 \( 1 - 237. iT - 5.06e4T^{2} \)
41 \( 1 - 195.T + 6.89e4T^{2} \)
43 \( 1 - 343. iT - 7.95e4T^{2} \)
47 \( 1 - 16.8iT - 1.03e5T^{2} \)
53 \( 1 + 346. iT - 1.48e5T^{2} \)
59 \( 1 + 135.T + 2.05e5T^{2} \)
61 \( 1 - 490.T + 2.26e5T^{2} \)
67 \( 1 - 477. iT - 3.00e5T^{2} \)
71 \( 1 - 45.2T + 3.57e5T^{2} \)
73 \( 1 + 100. iT - 3.89e5T^{2} \)
79 \( 1 + 880.T + 4.93e5T^{2} \)
83 \( 1 - 1.15e3iT - 5.71e5T^{2} \)
89 \( 1 + 619.T + 7.04e5T^{2} \)
97 \( 1 + 231. iT - 9.12e5T^{2} \)
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

−11.77536243395548785257976779449, −11.28707936677505099714370714250, −9.998698142794822974880169488861, −9.538748644707741001867558966944, −8.684800736064690741660805855922, −6.74619054596288731619438029360, −4.82347846601635208633376369084, −4.09598734150233839377106598696, −2.80809988348601858009916127570, −1.23678892508664336114345203982, 1.03829742912228971294156827835, 3.78630649336900059799317552286, 5.33710611293323790785216487191, 6.25072084857164076188229077362, 7.42581254420557521822978823064, 7.68162176055508215004421389016, 9.031264823009605862549773932906, 10.06055026914113642691433312932, 11.70564918613939372239503277867, 12.85454557550799925229938126165

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