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Properties

Label 2-3900-195.134-c0-0-1
Degree $2$
Conductor $3900$
Sign $0.435 - 0.900i$
Analytic cond. $1.94635$
Root an. cond. $1.39511$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (0.499 + 0.866i)9-s + (0.866 + 0.5i)13-s + (−1.5 + 0.866i)19-s + 0.999i·27-s + 1.73i·31-s + (0.499 + 0.866i)39-s + (1.73 − i)43-s + (0.5 − 0.866i)49-s − 1.73·57-s + (−0.5 − 0.866i)61-s + 1.73·73-s − 79-s + (−0.5 + 0.866i)81-s + (−0.866 + 1.49i)93-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (0.499 + 0.866i)9-s + (0.866 + 0.5i)13-s + (−1.5 + 0.866i)19-s + 0.999i·27-s + 1.73i·31-s + (0.499 + 0.866i)39-s + (1.73 − i)43-s + (0.5 − 0.866i)49-s − 1.73·57-s + (−0.5 − 0.866i)61-s + 1.73·73-s − 79-s + (−0.5 + 0.866i)81-s + (−0.866 + 1.49i)93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.435 - 0.900i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (3449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ 0.435 - 0.900i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.767355866\)
\(L(\frac12)\) \(\approx\) \(1.767355866\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 \)
13 \( 1 + (-0.866 - 0.5i)T \)
good7 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.73iT - T^{2} \)
37 \( 1 + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.5 + 0.866i)T^{2} \)
73 \( 1 - 1.73T + T^{2} \)
79 \( 1 + T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
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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

−8.661264276142602446437422293834, −8.351524413551270593406848485267, −7.40317154976049672688686595165, −6.64468014221597174181497579896, −5.81600619371402778365857783434, −4.85554067116649025387767333783, −4.03231212777214030345700187017, −3.51449201156394624839295351723, −2.40832455888900783305751058713, −1.56804790254986706114021683889, 0.948652372340011263077052331377, 2.19279596936946163358495792563, 2.84701478535461710262618426207, 3.92117154666021487542894792022, 4.45599700101033301715028446394, 5.81171124675168225810557109275, 6.32399483767103642117659951307, 7.16495357694549285213844468114, 7.902290504622424974867690784667, 8.437581457814318703182493781162

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