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Properties

Label 2-70e2-5.4-c1-0-45
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  − i·3-s + 2·9-s + 6·11-s − 2i·13-s − 6i·17-s + 8·19-s + 3i·23-s − 5i·27-s − 3·29-s − 2·31-s − 6i·33-s − 8i·37-s − 2·39-s + 3·41-s + 5i·43-s + ⋯
L(s)  = 1  − 0.577i·3-s + 0.666·9-s + 1.80·11-s − 0.554i·13-s − 1.45i·17-s + 1.83·19-s + 0.625i·23-s − 0.962i·27-s − 0.557·29-s − 0.359·31-s − 1.04i·33-s − 1.31i·37-s − 0.320·39-s + 0.468·41-s + 0.762i·43-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

\(L(1)\) \(\approx\) \(2.591210673\)
\(L(\frac12)\) \(\approx\) \(2.591210673\)
\(L(\frac{3}{2})\) 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 \( 1 + iT - 3T^{2} \)
11 \( 1 - 6T + 11T^{2} \)
13 \( 1 + 2iT - 13T^{2} \)
17 \( 1 + 6iT - 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + 2T + 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 5iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 12iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - T + 61T^{2} \)
67 \( 1 - 7iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 10iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 3iT - 83T^{2} \)
89 \( 1 + 3T + 89T^{2} \)
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

−7.86752332192373604831485677850, −7.28891948042993951860601591798, −6.96732521651290807865312985415, −5.97476631605938133342484026571, −5.33080307114743325718367476744, −4.33132987979400461158742265795, −3.60589671892736616441633664894, −2.69657897532409626969289104444, −1.46634521361575434593484116835, −0.870041381074443320868050133853, 1.16852305385555366048474044306, 1.86702851527899478090995427122, 3.42093321071377068114770700622, 3.82722816236990703944666985505, 4.56312248845335766050668530885, 5.39342702097849832509974697530, 6.37125555402403764607217372538, 6.81313793144135731210537748877, 7.64454205141315531833549156978, 8.515325812594922170917733545634

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