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Properties

Label 2-3528-7.2-c1-0-32
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  + (1.41 − 2.44i)5-s + (3 + 5.19i)11-s − 5.65·13-s + (0.707 + 1.22i)17-s + (2.12 − 3.67i)19-s + (2 − 3.46i)23-s + (−1.49 − 2.59i)25-s + 6·29-s + (−1.41 − 2.44i)31-s + (−1 + 1.73i)37-s + 1.41·41-s + 10·43-s + (−1.41 + 2.44i)47-s + (−1 − 1.73i)53-s + 16.9·55-s + ⋯
L(s)  = 1  + (0.632 − 1.09i)5-s + (0.904 + 1.56i)11-s − 1.56·13-s + (0.171 + 0.297i)17-s + (0.486 − 0.842i)19-s + (0.417 − 0.722i)23-s + (−0.299 − 0.519i)25-s + 1.11·29-s + (−0.254 − 0.439i)31-s + (−0.164 + 0.284i)37-s + 0.220·41-s + 1.52·43-s + (−0.206 + 0.357i)47-s + (−0.137 − 0.237i)53-s + 2.28·55-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

\(L(1)\) \(\approx\) \(2.164608352\)
\(L(\frac12)\) \(\approx\) \(2.164608352\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.65T + 13T^{2} \)
17 \( 1 + (-0.707 - 1.22i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.12 + 3.67i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (1.41 + 2.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 1.41T + 41T^{2} \)
43 \( 1 - 10T + 43T^{2} \)
47 \( 1 + (1.41 - 2.44i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1 + 1.73i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.707 - 1.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.24 + 7.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-4.94 - 8.57i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 1.41T + 83T^{2} \)
89 \( 1 + (2.12 - 3.67i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 12.7T + 97T^{2} \)
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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

−8.663269594959607514580130855569, −7.69124760077451187062367711420, −7.03197097044965247086626020091, −6.34460081096287533084252790136, −5.17024516401792180751999459980, −4.82369346904779190028842941449, −4.13396259752272635554790885722, −2.66951369751552201906510790091, −1.90240425958042922724645101571, −0.818427862425485980054811917848, 0.965834302068973342007783509797, 2.29962872027334549164651060311, 3.06016096326889716302629916130, 3.75428278706412503730207237697, 4.99219139399494770014565902653, 5.77113789086472526912067669573, 6.38274914801200633961910028049, 7.12680006927557319609787599541, 7.74722007394402567149590807576, 8.743280139463703507109080973827

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