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Properties

Label 2-3900-195.74-c0-0-0
Degree $2$
Conductor $3900$
Sign $-0.310 - 0.950i$
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 + (−0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)13-s + (1 + 1.73i)19-s − 0.999·21-s + 0.999i·27-s − 31-s + (−1.73 − i)37-s − 0.999·39-s + (0.866 − 0.5i)43-s + 1.99i·57-s + (0.5 + 0.866i)61-s + (−0.866 − 0.499i)63-s + (0.866 + 0.5i)67-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.866 + 0.5i)7-s + (0.499 + 0.866i)9-s + (−0.866 + 0.5i)13-s + (1 + 1.73i)19-s − 0.999·21-s + 0.999i·27-s − 31-s + (−1.73 − i)37-s − 0.999·39-s + (0.866 − 0.5i)43-s + 1.99i·57-s + (0.5 + 0.866i)61-s + (−0.866 − 0.499i)63-s + (0.866 + 0.5i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.310 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-0.310 - 0.950i$
Analytic conductor: \(1.94635\)
Root analytic conductor: \(1.39511\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3900} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3900,\ (\ :0),\ -0.310 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.357823950\)
\(L(\frac12)\) \(\approx\) \(1.357823950\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + T + T^{2} \)
37 \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 - 0.866i)T^{2} \)
73 \( 1 - iT - T^{2} \)
79 \( 1 - T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.061396006120643595452717729175, −8.201093418515172107729424458986, −7.45014743030896858306464652297, −6.87159116409247664205669054235, −5.70830185504119736654106332588, −5.22456062722578984995285376400, −4.02266624640152278932935022076, −3.52056204233690099996045639511, −2.60312525979828824668957178513, −1.75516384687044574321441458248, 0.65661728146322750035283375383, 2.03874594373791569994623177968, 3.03854603575041572333902208271, 3.44837392622569684206313922335, 4.60288350498604738757894019457, 5.41039796952086160572971188511, 6.56935716630438742025547788270, 7.04512639740825851707113012310, 7.57251036771599235443580585867, 8.407604563109455907218924759909

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