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Properties

Label 2-3528-21.5-c1-0-39
Degree $2$
Conductor $3528$
Sign $-0.999 - 0.0285i$
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  + (1.84 − 3.20i)5-s + (−2.95 + 1.70i)11-s − 5.22i·13-s + (−3.37 − 5.85i)17-s + (−1.87 − 1.08i)19-s + (5.40 + 3.12i)23-s + (−4.32 − 7.49i)25-s + 2.58i·29-s + (−9.05 + 5.22i)31-s + (−2 + 3.46i)37-s − 0.634·41-s − 6.48·43-s + (1.53 − 2.65i)47-s + (−2.23 + 1.29i)53-s + 12.6i·55-s + ⋯
L(s)  = 1  + (0.826 − 1.43i)5-s + (−0.891 + 0.514i)11-s − 1.44i·13-s + (−0.819 − 1.41i)17-s + (−0.430 − 0.248i)19-s + (1.12 + 0.650i)23-s + (−0.865 − 1.49i)25-s + 0.480i·29-s + (−1.62 + 0.938i)31-s + (−0.328 + 0.569i)37-s − 0.0990·41-s − 0.988·43-s + (0.223 − 0.386i)47-s + (−0.307 + 0.177i)53-s + 1.70i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0285i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0285i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.999 - 0.0285i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (1097, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.999 - 0.0285i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9554012272\)
\(L(\frac12)\) \(\approx\) \(0.9554012272\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 + 3.20i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.95 - 1.70i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.22iT - 13T^{2} \)
17 \( 1 + (3.37 + 5.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.87 + 1.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.40 - 3.12i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.58iT - 29T^{2} \)
31 \( 1 + (9.05 - 5.22i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (2 - 3.46i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.634T + 41T^{2} \)
43 \( 1 + 6.48T + 43T^{2} \)
47 \( 1 + (-1.53 + 2.65i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.23 - 1.29i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.828 + 1.43i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 7.41iT - 71T^{2} \)
73 \( 1 + (-0.776 + 0.448i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.82 + 11.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + (3.37 - 5.85i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.55iT - 97T^{2} \)
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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

−8.331840577809707923755501793235, −7.49597153707701452568676550705, −6.78176645455028573055602239943, −5.60770941213041607713366827074, −5.05577193575518862836725295398, −4.83847562412215794368012644845, −3.35942208025120528319334341047, −2.43663268431455960060523490946, −1.40646014430156779087060370291, −0.26064451854586496212635660462, 1.87807222035528587397981926723, 2.35606046370772230478807572695, 3.41075906948693274067970058821, 4.21587385881129520523589455823, 5.33848393980344857147551567827, 6.18268836418863776226600611053, 6.58994008586648163749084249936, 7.28923138082922975839526809103, 8.213760511086563273791164093965, 9.018484981518682002739389066815

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