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Properties

Label 2-3900-39.23-c0-0-1
Degree $2$
Conductor $3900$
Sign $0.0128 - 0.999i$
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  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)13-s + (1.5 + 0.866i)19-s − 0.999·27-s − 1.73i·31-s + (−0.499 + 0.866i)39-s + (−1 + 1.73i)43-s + (−0.5 − 0.866i)49-s + 1.73i·57-s + (−0.5 + 0.866i)61-s + 1.73i·73-s + 79-s + (−0.5 − 0.866i)81-s + (1.49 − 0.866i)93-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)13-s + (1.5 + 0.866i)19-s − 0.999·27-s − 1.73i·31-s + (−0.499 + 0.866i)39-s + (−1 + 1.73i)43-s + (−0.5 − 0.866i)49-s + 1.73i·57-s + (−0.5 + 0.866i)61-s + 1.73i·73-s + 79-s + (−0.5 − 0.866i)81-s + (1.49 − 0.866i)93-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ 0.0128 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.532562277\)
\(L(\frac12)\) \(\approx\) \(1.532562277\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 \)
13 \( 1 + (-0.5 - 0.866i)T \)
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 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
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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

−8.936104505466207044425334840690, −8.092904411573038579997029623664, −7.63759840631090970429980966923, −6.57233846261268552855943094587, −5.75705222044825545105670494208, −5.01130348146937065721501231862, −4.13916398893835078037280896904, −3.53879485948109605659106192558, −2.62274231547643790656431771770, −1.51756208510379654885689470223, 0.869259047356065948060619999923, 1.89750458733582890972048453101, 3.12168952979126617023341546592, 3.39501388641324630185687193573, 4.83253962134920458012084918242, 5.54238634308759343765891220069, 6.40271787862146294330076803019, 7.11843748694741079247570740826, 7.68221279566344749179762123010, 8.451204112908210830382541084940

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