Properties

Label 2-1725-5.4-c1-0-17
Degree $2$
Conductor $1725$
Sign $-0.447 - 0.894i$
Analytic cond. $13.7741$
Root an. cond. $3.71136$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·3-s + 2·4-s + 3i·7-s − 9-s − 4·11-s + 2i·12-s + 4·16-s + 3i·17-s + 8·19-s − 3·21-s + i·23-s i·27-s + 6i·28-s − 9·29-s − 5·31-s + ⋯
L(s)  = 1  + 0.577i·3-s + 4-s + 1.13i·7-s − 0.333·9-s − 1.20·11-s + 0.577i·12-s + 16-s + 0.727i·17-s + 1.83·19-s − 0.654·21-s + 0.208i·23-s − 0.192i·27-s + 1.13i·28-s − 1.67·29-s − 0.898·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1725 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1725\)    =    \(3 \cdot 5^{2} \cdot 23\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(13.7741\)
Root analytic conductor: \(3.71136\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1725} (1174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1725,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.814189703\)
\(L(\frac12)\) \(\approx\) \(1.814189703\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - iT \)
5 \( 1 \)
23 \( 1 - iT \)
good2 \( 1 - 2T^{2} \)
7 \( 1 - 3iT - 7T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 - 3iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
29 \( 1 + 9T + 29T^{2} \)
31 \( 1 + 5T + 31T^{2} \)
37 \( 1 - 9iT - 37T^{2} \)
41 \( 1 - 7T + 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 - 2iT - 47T^{2} \)
53 \( 1 - 13iT - 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 + 13iT - 67T^{2} \)
71 \( 1 + 13T + 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + 10iT - 97T^{2} \)
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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

−9.565279338826413960971259546226, −8.926532559019562022816944927405, −7.75770461691618372062245578173, −7.50090320118022509772177881688, −6.00536206012827112974817969990, −5.71754463173331945510285075426, −4.83003050549640084082884850282, −3.36615658012461619407368530439, −2.78268081366154765488181893942, −1.69207891475095067150857139261, 0.63611834226249173720756556560, 1.90833694920186829033799034347, 2.91937636166616606508810148931, 3.79911427582326073073003096685, 5.25993559060611595006717698258, 5.77971687657674291872529010312, 7.06205038912928566350026728631, 7.42584380228740967621447704656, 7.77060578566631172700113499912, 9.100249813399176596182929467737

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