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Properties

Label 2-3900-195.188-c0-0-0
Degree $2$
Conductor $3900$
Sign $0.861 + 0.507i$
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  + (−0.965 + 0.258i)3-s + (−0.258 − 0.448i)7-s + (0.866 − 0.499i)9-s + (−0.258 + 0.965i)13-s + (0.366 − 1.36i)19-s + (0.366 + 0.366i)21-s + (−0.707 + 0.707i)27-s + (−0.366 + 0.366i)31-s + (0.707 − 1.22i)37-s − i·39-s + (0.965 + 0.258i)43-s + (0.366 − 0.633i)49-s + 1.41i·57-s + (0.866 + 1.5i)61-s + (−0.448 − 0.258i)63-s + ⋯
L(s)  = 1  + (−0.965 + 0.258i)3-s + (−0.258 − 0.448i)7-s + (0.866 − 0.499i)9-s + (−0.258 + 0.965i)13-s + (0.366 − 1.36i)19-s + (0.366 + 0.366i)21-s + (−0.707 + 0.707i)27-s + (−0.366 + 0.366i)31-s + (0.707 − 1.22i)37-s − i·39-s + (0.965 + 0.258i)43-s + (0.366 − 0.633i)49-s + 1.41i·57-s + (0.866 + 1.5i)61-s + (−0.448 − 0.258i)63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.861 + 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.861 + 0.507i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (3893, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ 0.861 + 0.507i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8345352642\)
\(L(\frac12)\) \(\approx\) \(0.8345352642\)
\(L(1)\) 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 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
13 \( 1 + (0.258 - 0.965i)T \)
good7 \( 1 + (0.258 + 0.448i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.366 + 1.36i)T + (-0.866 - 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.366 - 0.366i)T - iT^{2} \)
37 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T^{2} \)
43 \( 1 + (-0.965 - 0.258i)T + (0.866 + 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + (-0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (1.67 + 0.965i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.866 - 0.5i)T^{2} \)
73 \( 1 + 0.517iT - T^{2} \)
79 \( 1 + 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.866 - 0.5i)T^{2} \)
97 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
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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

−8.914773737709013543706691724868, −7.44227917931910832178081754756, −7.18731148600499191653528472879, −6.34213165693327569796094835554, −5.66092365811885559406468537248, −4.71333144348040251571666189639, −4.25810565069313575753002503838, −3.24773680877911987238653033203, −2.00934984551683151462858834097, −0.67570012275476451581148897410, 1.01174257164518030990010773275, 2.19843649250859811627216079508, 3.26836313947569916576714059218, 4.25869907363972179998603304327, 5.17581053702726118064489044922, 5.81067859943877937533124710840, 6.27825716393412115733096845333, 7.29891747923153886791090904336, 7.82356828345524259930116142711, 8.599293870253778315374182076355

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