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Properties

Label 2-3528-7.2-c1-0-22
Degree $2$
Conductor $3528$
Sign $0.605 - 0.795i$
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  + (−2 + 3.46i)5-s + 3·13-s + (2 + 3.46i)17-s + (3.5 − 6.06i)19-s + (2 − 3.46i)23-s + (−5.49 − 9.52i)25-s + 8·29-s + (−2.5 − 4.33i)31-s + (−1.5 + 2.59i)37-s + 8·41-s + 11·43-s + (−2 + 3.46i)47-s + (2 + 3.46i)53-s + (−6 − 10.3i)59-s + (−1 + 1.73i)61-s + ⋯
L(s)  = 1  + (−0.894 + 1.54i)5-s + 0.832·13-s + (0.485 + 0.840i)17-s + (0.802 − 1.39i)19-s + (0.417 − 0.722i)23-s + (−1.09 − 1.90i)25-s + 1.48·29-s + (−0.449 − 0.777i)31-s + (−0.246 + 0.427i)37-s + 1.24·41-s + 1.67·43-s + (−0.291 + 0.505i)47-s + (0.274 + 0.475i)53-s + (−0.781 − 1.35i)59-s + (−0.128 + 0.221i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.605 - 0.795i$
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: \((2,\ 3528,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.699605558\)
\(L(\frac12)\) \(\approx\) \(1.699605558\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - 3.46i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 3T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 8T + 41T^{2} \)
43 \( 1 - 11T + 43T^{2} \)
47 \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2 - 3.46i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (4 - 6.92i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−8.559561959863017829709410804226, −7.78087061403301152720187519574, −7.26360145984578740217757323890, −6.46938486397854112002955414682, −5.97253008847126941210587264287, −4.69248046830280884918663871053, −3.91336809577942005458918695130, −3.10624492650664100747341317179, −2.50530360609800752838603368437, −0.857190845935932183204033553027, 0.77852974664758079529937657563, 1.47193992646419057075097411795, 3.10590490004609489005453155726, 3.86966211393064922194364254045, 4.62288230374164225907320541892, 5.39336738003913672917276657487, 5.98531443713491862603752124567, 7.32209167454931939056938177301, 7.72223129594299051355931446332, 8.529369687634611568325823557103

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