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Properties

Label 2-3564-9.7-c1-0-18
Degree $2$
Conductor $3564$
Sign $0.173 - 0.984i$
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  + (1.5 + 2.59i)5-s + (−1 + 1.73i)7-s + (0.5 − 0.866i)11-s + (2 + 3.46i)13-s + 6·17-s + 8·19-s + (1.5 + 2.59i)23-s + (−2 + 3.46i)25-s + (−2.5 − 4.33i)31-s − 6·35-s − 37-s + (5 − 8.66i)43-s + (1.50 + 2.59i)49-s − 6·53-s + 3·55-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)5-s + (−0.377 + 0.654i)7-s + (0.150 − 0.261i)11-s + (0.554 + 0.960i)13-s + 1.45·17-s + 1.83·19-s + (0.312 + 0.541i)23-s + (−0.400 + 0.692i)25-s + (−0.449 − 0.777i)31-s − 1.01·35-s − 0.164·37-s + (0.762 − 1.32i)43-s + (0.214 + 0.371i)49-s − 0.824·53-s + 0.404·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3564 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.173 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3564\)    =    \(2^{2} \cdot 3^{4} \cdot 11\)
Sign: $0.173 - 0.984i$
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: \((2,\ 3564,\ (\ :1/2),\ 0.173 - 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.413938862\)
\(L(\frac12)\) \(\approx\) \(2.413938862\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 - 2.59i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
13 \( 1 + (-2 - 3.46i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 - 8T + 19T^{2} \)
23 \( 1 + (-1.5 - 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 15T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + (1 - 1.73i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 + (-3.5 + 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.926434128680413900189421467074, −7.82052508460738019841530154681, −7.19946560308084316252250869096, −6.44681148710493573829506001108, −5.75996178414549906868520726242, −5.26751563340621426021870344995, −3.77163562821614722168704124791, −3.20507799836785075979211904319, −2.36201874265974489515788319298, −1.25766853972179837158493037725, 0.877612638413046658514769773456, 1.36858952174135289762131848096, 2.95393748634520526760662379516, 3.62825075623247974172779662330, 4.74873982556941865839999898562, 5.41134417198465225247594992463, 5.91704297201557135030081110053, 7.00926373562777410683727937119, 7.71284647679952835194801016919, 8.382221206693734516570820314752

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