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Properties

Label 2-70e2-5.4-c1-0-58
Degree $2$
Conductor $4900$
Sign $-0.447 - 0.894i$
Analytic cond. $39.1266$
Root an. cond. $6.25513$
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  − 3i·3-s − 6·9-s − 2·11-s − 6i·13-s − 2i·17-s − 9i·23-s + 9i·27-s − 3·29-s + 2·31-s + 6i·33-s − 8i·37-s − 18·39-s + 5·41-s + i·43-s − 8i·47-s + ⋯
L(s)  = 1  − 1.73i·3-s − 2·9-s − 0.603·11-s − 1.66i·13-s − 0.485i·17-s − 1.87i·23-s + 1.73i·27-s − 0.557·29-s + 0.359·31-s + 1.04i·33-s − 1.31i·37-s − 2.88·39-s + 0.780·41-s + 0.152i·43-s − 1.16i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4900\)    =    \(2^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(39.1266\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4900} (2549, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4900,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.059249978\)
\(L(\frac12)\) \(\approx\) \(1.059249978\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + 3iT - 3T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 6iT - 13T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 9iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 5T + 41T^{2} \)
43 \( 1 - iT - 43T^{2} \)
47 \( 1 + 8iT - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 7T + 61T^{2} \)
67 \( 1 - 3iT - 67T^{2} \)
71 \( 1 - 8T + 71T^{2} \)
73 \( 1 - 14iT - 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + iT - 83T^{2} \)
89 \( 1 + 13T + 89T^{2} \)
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

−7.75819461750138587573321806343, −7.11633872769919272074151889212, −6.48257535512681288714544784869, −5.64733386520496298938240457734, −5.20094445439128630483935361484, −3.86902306078908157210361053440, −2.57398002177700183351243545641, −2.48718570798707084033580842034, −1.00397152571148135835919303822, −0.32058553311134252531728694303, 1.68351657563366189079604723233, 2.83743416889672640116737305572, 3.70111564415404097309701575085, 4.24254958085369080754719515782, 4.97783158302393580326857325548, 5.61406359241653358984327465588, 6.42463364161927056867041349096, 7.38036159906017304798779246382, 8.244832434877401667982860938815, 8.927606384832135505937107537569

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