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Properties

Label 2-3564-9.4-c1-0-9
Degree $2$
Conductor $3564$
Sign $0.342 - 0.939i$
Analytic cond. $28.4586$
Root an. cond. $5.33466$
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  + (−0.327 + 0.567i)5-s + (−1.34 − 2.32i)7-s + (0.5 + 0.866i)11-s + (−0.0382 + 0.0662i)13-s − 1.89·17-s − 8.09·19-s + (3.60 − 6.23i)23-s + (2.28 + 3.95i)25-s + (0.157 + 0.272i)29-s + (−3.86 + 6.68i)31-s + 1.75·35-s + 6.04·37-s + (4.59 − 7.95i)41-s + (4.72 + 8.18i)43-s + (6.37 + 11.0i)47-s + ⋯
L(s)  = 1  + (−0.146 + 0.253i)5-s + (−0.506 − 0.877i)7-s + (0.150 + 0.261i)11-s + (−0.0106 + 0.0183i)13-s − 0.458·17-s − 1.85·19-s + (0.751 − 1.30i)23-s + (0.457 + 0.791i)25-s + (0.0292 + 0.0506i)29-s + (−0.693 + 1.20i)31-s + 0.297·35-s + 0.993·37-s + (0.717 − 1.24i)41-s + (0.721 + 1.24i)43-s + (0.929 + 1.60i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $0.342 - 0.939i$
Analytic conductor: \(28.4586\)
Root analytic conductor: \(5.33466\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3564} (2377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.118566894\)
\(L(\frac12)\) \(\approx\) \(1.118566894\)
\(L(\frac{3}{2})\) 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 \( 1 \)
11 \( 1 + (-0.5 - 0.866i)T \)
good5 \( 1 + (0.327 - 0.567i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.34 + 2.32i)T + (-3.5 + 6.06i)T^{2} \)
13 \( 1 + (0.0382 - 0.0662i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 1.89T + 17T^{2} \)
19 \( 1 + 8.09T + 19T^{2} \)
23 \( 1 + (-3.60 + 6.23i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.157 - 0.272i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.86 - 6.68i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 6.04T + 37T^{2} \)
41 \( 1 + (-4.59 + 7.95i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.72 - 8.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.37 - 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 9.59T + 53T^{2} \)
59 \( 1 + (4.93 - 8.55i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.47 - 6.02i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.00872 - 0.0151i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 8.54T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 + (-3.64 - 6.31i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.29 + 9.17i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 12.6T + 89T^{2} \)
97 \( 1 + (3.66 + 6.34i)T + (-48.5 + 84.0i)T^{2} \)
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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

−8.875262714547586070925828021092, −7.86356598899655540748258757423, −7.11030304571450674846696187625, −6.61373720531286644819780026876, −5.89500199570099352635343469144, −4.58657113645126830666886297424, −4.24153073691152460528342198012, −3.18607025434360716677954568912, −2.31078013724778631616135690713, −0.968475168096172642896231296826, 0.39077292849837075034918566846, 1.95358355371524707111770799487, 2.72738477922182381202637027350, 3.79318024420768047291231855763, 4.53656660587171450087550200617, 5.51305164486850240182419544555, 6.17568040714487095059814276298, 6.79021391306178449965564496460, 7.82305376220482878920034045052, 8.461248750165914917405697898399

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