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Properties

Label 2-175-5.4-c3-0-11
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

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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 \( 1 - 231. iT - 9.12e5T^{2} \)
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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

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

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