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Properties

Label 2-3564-9.7-c1-0-31
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  + (0.992 + 1.71i)5-s + (0.287 − 0.498i)7-s + (0.5 − 0.866i)11-s + (−1.35 − 2.35i)13-s − 5.85·17-s + 7.40·19-s + (−1.63 − 2.83i)23-s + (0.528 − 0.914i)25-s + (2.00 − 3.47i)29-s + (−2.36 − 4.08i)31-s + 1.14·35-s − 8.38·37-s + (−5.57 − 9.65i)41-s + (0.458 − 0.794i)43-s + (0.0712 − 0.123i)47-s + ⋯
L(s)  = 1  + (0.444 + 0.769i)5-s + (0.108 − 0.188i)7-s + (0.150 − 0.261i)11-s + (−0.376 − 0.652i)13-s − 1.41·17-s + 1.69·19-s + (−0.340 − 0.590i)23-s + (0.105 − 0.182i)25-s + (0.373 − 0.646i)29-s + (−0.423 − 0.734i)31-s + 0.193·35-s − 1.37·37-s + (−0.870 − 1.50i)41-s + (0.0699 − 0.121i)43-s + (0.0103 − 0.0180i)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: $\chi_{3564} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3564,\ (\ :1/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.574733318\)
\(L(\frac12)\) \(\approx\) \(1.574733318\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.992 - 1.71i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-0.287 + 0.498i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (1.35 + 2.35i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 5.85T + 17T^{2} \)
19 \( 1 - 7.40T + 19T^{2} \)
23 \( 1 + (1.63 + 2.83i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.00 + 3.47i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.36 + 4.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 8.38T + 37T^{2} \)
41 \( 1 + (5.57 + 9.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.458 + 0.794i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.0712 + 0.123i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 3.52T + 53T^{2} \)
59 \( 1 + (-6.19 - 10.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.94 + 5.10i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.93 + 8.55i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.86T + 71T^{2} \)
73 \( 1 - 9.04T + 73T^{2} \)
79 \( 1 + (-4.74 + 8.21i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.98 + 5.16i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6.94T + 89T^{2} \)
97 \( 1 + (-9.45 + 16.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.484741238518832925286408800485, −7.48409458106919629552889920339, −7.02396431558705268871444726348, −6.18802798521427997313562746087, −5.49751412392946797138710994102, −4.61533680786955125107204079402, −3.63649033853603718046736900076, −2.77161094241653890745660102882, −1.97766951426766300949793994180, −0.47325237376789627085848592343, 1.24622762675949542432651699775, 2.03003673434101113213168695723, 3.18336926832485515223346619370, 4.17960282116048465332923090212, 5.18182952856729100997670658125, 5.32455169445854064253468317882, 6.69407598584995228853046416841, 7.02629228078375937368805087588, 8.090899516382298923666272543395, 8.849454981708426513326487486045

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