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Properties

Label 2-3528-7.2-c1-0-30
Degree $2$
Conductor $3528$
Sign $0.827 + 0.561i$
Analytic cond. $28.1712$
Root an. cond. $5.30765$
Motivic weight $1$
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 − 1.22i)5-s − 1.41·13-s + (−0.707 − 1.22i)17-s + (−2 + 3.46i)23-s + (1.50 + 2.59i)25-s + 6·29-s + (2.82 + 4.89i)31-s + (2 − 3.46i)37-s + 7.07·41-s + 4·43-s + (5.65 − 9.79i)47-s + (−2 − 3.46i)53-s + (−2.82 − 4.89i)59-s + (2.12 − 3.67i)61-s + (−1.00 + 1.73i)65-s + ⋯
L(s)  = 1  + (0.316 − 0.547i)5-s − 0.392·13-s + (−0.171 − 0.297i)17-s + (−0.417 + 0.722i)23-s + (0.300 + 0.519i)25-s + 1.11·29-s + (0.508 + 0.879i)31-s + (0.328 − 0.569i)37-s + 1.10·41-s + 0.609·43-s + (0.825 − 1.42i)47-s + (−0.274 − 0.475i)53-s + (−0.368 − 0.637i)59-s + (0.271 − 0.470i)61-s + (−0.124 + 0.214i)65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.827 + 0.561i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (3313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.920004350\)
\(L(\frac12)\) \(\approx\) \(1.920004350\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + (-0.707 + 1.22i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (0.707 + 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-2.82 - 4.89i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2 + 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.07T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-5.65 + 9.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.82 + 4.89i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.12 + 3.67i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (3.53 + 6.12i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 + (-6.36 + 11.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−8.564499442854433339548822219990, −7.78213480230713943947099081907, −7.04310919169011911185091049256, −6.24424931291055014610273215391, −5.37502384962556578246960770968, −4.81196676248911064419850926058, −3.88185155922298331459972579758, −2.86861767911408905010837352001, −1.88481234374815161419701063678, −0.73763411856540830197876763652, 0.926092722630014605111445645099, 2.36675833024072717194424064416, 2.83885649181238646434063600287, 4.14168364565290695408132210556, 4.67878866564645690286248081441, 5.88976333395735968151038902236, 6.30211889448959292826186510061, 7.13503379597963443294311077008, 7.908428936445971607070469845461, 8.581461037077837911391877267621

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