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Properties

Label 2-3528-24.5-c0-0-3
Degree $2$
Conductor $3528$
Sign $-0.985 - 0.169i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s − 1.41·11-s − 1.00·16-s + (1.00 + 1.00i)22-s − 1.41i·23-s − 25-s − 1.41·29-s + (0.707 + 0.707i)32-s + 2i·37-s − 2i·43-s − 1.41i·44-s + (−1.00 + 1.00i)46-s + (0.707 + 0.707i)50-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s − 1.41·11-s − 1.00·16-s + (1.00 + 1.00i)22-s − 1.41i·23-s − 25-s − 1.41·29-s + (0.707 + 0.707i)32-s + 2i·37-s − 2i·43-s − 1.41i·44-s + (−1.00 + 1.00i)46-s + (0.707 + 0.707i)50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(1.76070\)
Root analytic conductor: \(1.32691\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :0),\ -0.985 - 0.169i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1905053094\)
\(L(\frac12)\) \(\approx\) \(0.1905053094\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + T^{2} \)
11 \( 1 + 1.41T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41iT - T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2T + T^{2} \)
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

−8.378538841430262635017221725683, −7.82975588358122350536019560752, −7.15820660341272734156273639287, −6.21607743308674196319828175543, −5.20559925027765817120152208585, −4.39548234458146003576983714281, −3.41054478746214282874386287356, −2.59043182858815207006197770108, −1.75441314570331899574368621344, −0.13299040740965585101162682182, 1.57325743811055333423216118893, 2.55334368382664277380309793610, 3.78433654928380714870820229851, 4.86566617807112408505781935175, 5.62172937055521596787328557526, 6.03956018819755531990388225411, 7.28488622847550263371532109516, 7.60579720839669954640074633674, 8.214374287766958627495451252801, 9.184631271161157954234269190776

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