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Properties

Label 2-3528-8.3-c0-0-3
Degree $2$
Conductor $3528$
Sign $1$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s + 8-s + 16-s − 1.41·17-s + 1.41·19-s + 25-s + 32-s − 1.41·34-s + 1.41·38-s + 1.41·41-s + 50-s − 1.41·59-s + 64-s − 2·67-s − 1.41·68-s − 1.41·73-s + 1.41·76-s + 1.41·82-s − 1.41·83-s + 1.41·89-s + 1.41·97-s + 100-s − 2·107-s − 1.41·118-s + ⋯
L(s)  = 1  + 2-s + 4-s + 8-s + 16-s − 1.41·17-s + 1.41·19-s + 25-s + 32-s − 1.41·34-s + 1.41·38-s + 1.41·41-s + 50-s − 1.41·59-s + 64-s − 2·67-s − 1.41·68-s − 1.41·73-s + 1.41·76-s + 1.41·82-s − 1.41·83-s + 1.41·89-s + 1.41·97-s + 100-s − 2·107-s − 1.41·118-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (883, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.548505860\)
\(L(\frac12)\) \(\approx\) \(2.548505860\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 - 1.41T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41T + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.41T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 1.41T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + 1.41T + T^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.774029967312598213741477833416, −7.71170454377112592635782080580, −7.19305344069580450103399159983, −6.39339894044203444175283931800, −5.71632295172605145089621620672, −4.83019438946390675348486425956, −4.27995003549594251522400254868, −3.22730490348649438914891577355, −2.54020018782380841957006951260, −1.36467350816363020245700921873, 1.36467350816363020245700921873, 2.54020018782380841957006951260, 3.22730490348649438914891577355, 4.27995003549594251522400254868, 4.83019438946390675348486425956, 5.71632295172605145089621620672, 6.39339894044203444175283931800, 7.19305344069580450103399159983, 7.71170454377112592635782080580, 8.774029967312598213741477833416

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