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Properties

Label 2-3636-404.403-c0-0-13
Degree $2$
Conductor $3636$
Sign $i$
Analytic cond. $1.81460$
Root an. cond. $1.34707$
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  + i·2-s − 4-s − i·8-s − 2·13-s + 16-s − 25-s − 2i·26-s − 2i·29-s + i·32-s − 2·37-s + 2i·41-s − 49-s − i·50-s + 2·52-s − 2i·53-s + ⋯
L(s)  = 1  + i·2-s − 4-s − i·8-s − 2·13-s + 16-s − 25-s − 2i·26-s − 2i·29-s + i·32-s − 2·37-s + 2i·41-s − 49-s − i·50-s + 2·52-s − 2i·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3636 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3636 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3636\)    =    \(2^{2} \cdot 3^{2} \cdot 101\)
Sign: $i$
Analytic conductor: \(1.81460\)
Root analytic conductor: \(1.34707\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3636} (2827, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3636,\ (\ :0),\ i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1928754186\)
\(L(\frac12)\) \(\approx\) \(0.1928754186\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
101 \( 1 + iT \)
good5 \( 1 + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + 2T + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 - 2iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 2iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 2iT - T^{2} \)
97 \( 1 + 2T + T^{2} \)
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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

−8.218048049477679517395724602175, −7.83279802438047245073005477412, −7.05333106861163949742627532216, −6.42624556740142219679814078941, −5.52944010839176504349253421017, −4.87063690459201503845582003090, −4.19244924111953130542612495856, −3.11889178135142118311627125424, −1.95584084454801235834899658617, −0.10567147103543609202704403422, 1.60462028610666684595289769390, 2.45979501622897223019919294544, 3.31882994529198879414897090373, 4.20222501980008675275121166900, 5.11507318031558143287061376568, 5.48872061904387636771888305966, 6.88520841286645166730571199848, 7.46371095511333417894559811304, 8.334785541660182331911146706708, 9.153413059435556510894384282865

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