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Properties

Label 2-3528-7.2-c1-0-2
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  + (0.292 − 0.507i)5-s + (−0.414 − 0.717i)11-s − 1.41·13-s + (1.12 + 1.94i)17-s + (−3.41 + 5.91i)19-s + (2.41 − 4.18i)23-s + (2.32 + 4.03i)25-s − 8.48·29-s + (−2.58 − 4.47i)31-s + (−0.828 + 1.43i)37-s + 0.585·41-s − 8·43-s + (3.41 − 5.91i)47-s + (−6.65 − 11.5i)53-s − 0.485·55-s + ⋯
L(s)  = 1  + (0.130 − 0.226i)5-s + (−0.124 − 0.216i)11-s − 0.392·13-s + (0.271 + 0.471i)17-s + (−0.783 + 1.35i)19-s + (0.503 − 0.871i)23-s + (0.465 + 0.806i)25-s − 1.57·29-s + (−0.464 − 0.804i)31-s + (−0.136 + 0.235i)37-s + 0.0914·41-s − 1.21·43-s + (0.498 − 0.862i)47-s + (−0.914 − 1.58i)53-s − 0.0654·55-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

\(L(1)\) \(\approx\) \(0.4592098389\)
\(L(\frac12)\) \(\approx\) \(0.4592098389\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 + 0.507i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.414 + 0.717i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (-1.12 - 1.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.41 - 5.91i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2.41 + 4.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 8.48T + 29T^{2} \)
31 \( 1 + (2.58 + 4.47i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.828 - 1.43i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 0.585T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + (-3.41 + 5.91i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.65 + 11.5i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.58 - 4.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.94 - 12.0i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 0.828T + 71T^{2} \)
73 \( 1 + (-5.53 - 9.58i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.17 + 2.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (5.36 - 9.29i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 7.75T + 97T^{2} \)
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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

−8.738792792053096010539981368864, −8.239193570728277037213856642323, −7.40478168141134967300672925812, −6.66595006580714815248313715420, −5.75203775170795257455618652982, −5.24779595730172847428330341357, −4.16515756697189525985878739683, −3.50003047952040729549481904983, −2.34061335176742060505862135045, −1.41337219134422307120651697973, 0.13042985722133037087058254077, 1.64773071226144225704188001693, 2.65581266793225596130803943732, 3.43719571529364535493461284091, 4.58132737743965850840098578859, 5.12800775821086017063049200304, 6.06793362977773588065922454032, 6.90852219458131441659462253870, 7.40397731253128826193476066389, 8.244586165845846944891851231069

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