Properties

Label 2-3528-7.4-c1-0-8
Degree $2$
Conductor $3528$
Sign $0.198 - 0.980i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 - 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3528\)    =    \(2^{3} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.198 - 0.980i$
Analytic conductor: \(28.1712\)
Root analytic conductor: \(5.30765\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ 0.198 - 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.412055727\)
\(L(\frac12)\) \(\approx\) \(1.412055727\)
\(L(\frac{3}{2})\) 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

−8.711487342453584521636613691061, −7.929452185800570131850925456119, −7.42037760773212568351584695610, −6.42990204196634117883741559613, −5.72657192472292153422520739446, −4.99805489321598079160034974661, −3.98717871491676575225066621139, −3.38467182395386220156677602229, −2.14422576285114333819094969760, −1.16090453363432213960970350821, 0.45681719664315395911651944647, 1.77842614562926649478975705782, 2.98574639786145607685270553694, 3.49602136396230273029402932443, 4.83082456179263432827631213203, 5.14343501180586336268079560115, 6.35914030104591421842986402305, 6.88895515008303476323727363028, 7.61103865340633693302752919069, 8.467834758900601923805625821551

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