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Properties

Label 2-3648-456.227-c0-0-10
Degree $2$
Conductor $3648$
Sign $-0.707 + 0.707i$
Analytic cond. $1.82058$
Root an. cond. $1.34929$
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·3-s − 2i·7-s − 9-s − 2·13-s + i·19-s + 2·21-s − 25-s − i·27-s − 2·37-s − 2i·39-s − 3·49-s − 57-s + 2i·63-s − 2i·67-s − 2·73-s + ⋯
L(s)  = 1  + i·3-s − 2i·7-s − 9-s − 2·13-s + i·19-s + 2·21-s − 25-s − i·27-s − 2·37-s − 2i·39-s − 3·49-s − 57-s + 2i·63-s − 2i·67-s − 2·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3648\)    =    \(2^{6} \cdot 3 \cdot 19\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(1.82058\)
Root analytic conductor: \(1.34929\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3648} (1823, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3648,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2654996290\)
\(L(\frac12)\) \(\approx\) \(0.2654996290\)
\(L(1)\) 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 \( 1 - iT \)
19 \( 1 - iT \)
good5 \( 1 + T^{2} \)
7 \( 1 + 2iT - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 2T + T^{2} \)
17 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + 2T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 2T + T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.397616613300510958810890144980, −7.57112107805982369931097124450, −7.20666483295733535601620308319, −6.18285426648292667816863850567, −5.14202871952432946335314408730, −4.57580098256248156559917601791, −3.85374591264172394393880472894, −3.18836422865853217243472192008, −1.85718657363563900003094784587, −0.13665952458515185193401194843, 1.88363155899400568683412130301, 2.44727600902175491758915747791, 3.11186288382397028605164306127, 4.74078109174625150880051876284, 5.39625025398198863432201029833, 5.93511621922291648509305409728, 6.91014600374975325299612411113, 7.40526699913754225991920644846, 8.332688429815625284195337675794, 8.866996967088085816216547360174

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