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Properties

Label 2-3528-504.403-c0-0-4
Degree $2$
Conductor $3528$
Sign $0.415 - 0.909i$
Analytic cond. $1.76070$
Root an. cond. $1.32691$
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  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)6-s + 8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + 16-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)22-s + (0.5 + 0.866i)24-s + (−0.5 − 0.866i)25-s + ⋯
L(s)  = 1  + 2-s + (0.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)6-s + 8-s + (−0.499 + 0.866i)9-s + (0.5 + 0.866i)11-s + (0.5 + 0.866i)12-s + 16-s + (−0.5 + 0.866i)17-s + (−0.499 + 0.866i)18-s + (−0.5 − 0.866i)19-s + (0.5 + 0.866i)22-s + (0.5 + 0.866i)24-s + (−0.5 − 0.866i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.415 - 0.909i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.415 - 0.909i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (2419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ 0.415 - 0.909i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.891337144\)
\(L(\frac12)\) \(\approx\) \(2.891337144\)
\(L(1)\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.868320329685531293846956959772, −8.167578558847364501277142845723, −7.26686065516604024798251323788, −6.56741018628835899063694737414, −5.72036448764215642598741643293, −4.85488169532320179437857847262, −4.21831153950315463935850209968, −3.70429763108106388955683202623, −2.55812812276892009212820654145, −1.91556826351457052678913916276, 1.25223699249503463699553132193, 2.22635946106678392607468345056, 3.15085597277977048280272391480, 3.77282126933970821665267478568, 4.77685260662310705366140561125, 5.82980397746158007887450123493, 6.26367210459782307283985806503, 7.05462246201832031123260113220, 7.70676547718515562990107387411, 8.425826204894481936475446853936

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