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Properties

Label 2-3528-7.4-c1-0-39
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  + (−0.707 − 1.22i)5-s − 1.41·13-s + (0.707 − 1.22i)17-s + (2 + 3.46i)23-s + (1.50 − 2.59i)25-s − 6·29-s + (2.82 − 4.89i)31-s + (2 + 3.46i)37-s − 7.07·41-s + 4·43-s + (−5.65 − 9.79i)47-s + (2 − 3.46i)53-s + (2.82 − 4.89i)59-s + (2.12 + 3.67i)61-s + (1.00 + 1.73i)65-s + ⋯
L(s)  = 1  + (−0.316 − 0.547i)5-s − 0.392·13-s + (0.171 − 0.297i)17-s + (0.417 + 0.722i)23-s + (0.300 − 0.519i)25-s − 1.11·29-s + (0.508 − 0.879i)31-s + (0.328 + 0.569i)37-s − 1.10·41-s + 0.609·43-s + (−0.825 − 1.42i)47-s + (0.274 − 0.475i)53-s + (0.368 − 0.637i)59-s + (0.271 + 0.470i)61-s + (0.124 + 0.214i)65-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: $\chi_{3528} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3528,\ (\ :1/2),\ -0.827 + 0.561i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7688240261\)
\(L(\frac12)\) \(\approx\) \(0.7688240261\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.707 + 1.22i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.41T + 13T^{2} \)
17 \( 1 + (-0.707 + 1.22i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 + (-2.82 + 4.89i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 7.07T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (5.65 + 9.79i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.82 + 4.89i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.12 - 3.67i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (3.53 - 6.12i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 5.65T + 83T^{2} \)
89 \( 1 + (6.36 + 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 4.24T + 97T^{2} \)
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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

−8.349819795490596145634881167545, −7.51661592017484844779865357513, −6.93463216733873843059108693052, −5.92757556793308138554168758479, −5.19128089025752644497196692262, −4.46975333743737335792410808886, −3.61611702745574929849052638397, −2.63943209477540962930251172861, −1.49949994925692549872666866226, −0.23475173683378669180953926978, 1.34331493269820029966047917937, 2.57390793772450706760582085016, 3.31780367627722090858896433183, 4.23016332936894327691181618317, 5.07117458258182234703659394594, 5.91950646175083933977134892103, 6.76826027295107853890609088802, 7.33989884629984656152625243090, 8.059976448776608069031880305510, 8.874248682464346742414468434691

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