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Properties

Label 2-60e2-20.19-c0-0-5
Degree $2$
Conductor $3600$
Sign $0.834 + 0.550i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + 1.73·7-s − i·13-s − 1.73i·19-s − 1.73i·31-s + 2i·37-s − 1.73·43-s + 1.99·49-s − 61-s + 1.73·67-s + 2i·73-s − 1.73i·91-s + i·97-s − 109-s + ⋯
L(s)  = 1  + 1.73·7-s − i·13-s − 1.73i·19-s − 1.73i·31-s + 2i·37-s − 1.73·43-s + 1.99·49-s − 61-s + 1.73·67-s + 2i·73-s − 1.73i·91-s + i·97-s − 109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.834 + 0.550i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (1999, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.834 + 0.550i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.549653291\)
\(L(\frac12)\) \(\approx\) \(1.549653291\)
\(L(1)\) 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 \( 1 \)
5 \( 1 \)
good7 \( 1 - 1.73T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + iT - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + 1.73iT - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + 1.73iT - T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T + T^{2} \)
67 \( 1 - 1.73T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - iT - T^{2} \)
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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

−8.316039630159081937960184646316, −8.168450442229496865087168223032, −7.29439348410149312462414718646, −6.49521666378186052499218624138, −5.41496896244732757969989228173, −4.93779785185028632651561055680, −4.23025414926231834817358627162, −3.02057886471873872846792550654, −2.13633586337020783447403295665, −1.00332963900334483612024685842, 1.52453635648217505641684773312, 1.98355615760606567024379793763, 3.44735647957906007088366187053, 4.27178103654807156819458875344, 5.00001949153029499951211044986, 5.66943572544108749088877697102, 6.64126622485085421839863661979, 7.44850665029466781298473075189, 8.102856666018966157536094737223, 8.654442497532173379236243928507

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