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Properties

Label 2-2e5-8.3-c20-0-11
Degree $2$
Conductor $32$
Sign $0.810 - 0.586i$
Analytic cond. $81.1244$
Root an. cond. $9.00690$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.03e5·3-s − 1.04e7i·5-s + 3.68e8i·7-s + 7.17e9·9-s + 3.35e10·11-s + 1.95e11i·13-s − 1.07e12i·15-s − 1.59e11·17-s − 4.90e12·19-s + 3.80e13i·21-s + 3.43e13i·23-s − 1.34e13·25-s + 3.81e14·27-s + 1.07e14i·29-s − 3.09e13i·31-s + ⋯
L(s)  = 1  + 1.74·3-s − 1.06i·5-s + 1.30i·7-s + 2.05·9-s + 1.29·11-s + 1.41i·13-s − 1.86i·15-s − 0.0793·17-s − 0.799·19-s + 2.27i·21-s + 0.829i·23-s − 0.140·25-s + 1.85·27-s + 0.255i·29-s − 0.0378i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.810 - 0.586i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.810 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32\)    =    \(2^{5}\)
Sign: $0.810 - 0.586i$
Analytic conductor: \(81.1244\)
Root analytic conductor: \(9.00690\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{32} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 32,\ (\ :10),\ 0.810 - 0.586i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(4.677469451\)
\(L(\frac12)\) \(\approx\) \(4.677469451\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
good3 \( 1 - 1.03e5T + 3.48e9T^{2} \)
5 \( 1 + 1.04e7iT - 9.53e13T^{2} \)
7 \( 1 - 3.68e8iT - 7.97e16T^{2} \)
11 \( 1 - 3.35e10T + 6.72e20T^{2} \)
13 \( 1 - 1.95e11iT - 1.90e22T^{2} \)
17 \( 1 + 1.59e11T + 4.06e24T^{2} \)
19 \( 1 + 4.90e12T + 3.75e25T^{2} \)
23 \( 1 - 3.43e13iT - 1.71e27T^{2} \)
29 \( 1 - 1.07e14iT - 1.76e29T^{2} \)
31 \( 1 + 3.09e13iT - 6.71e29T^{2} \)
37 \( 1 - 4.26e15iT - 2.31e31T^{2} \)
41 \( 1 + 3.78e15T + 1.80e32T^{2} \)
43 \( 1 - 2.30e15T + 4.67e32T^{2} \)
47 \( 1 - 3.45e16iT - 2.76e33T^{2} \)
53 \( 1 + 3.51e16iT - 3.05e34T^{2} \)
59 \( 1 + 2.54e17T + 2.61e35T^{2} \)
61 \( 1 + 1.01e18iT - 5.08e35T^{2} \)
67 \( 1 - 2.98e18T + 3.32e36T^{2} \)
71 \( 1 + 8.39e17iT - 1.05e37T^{2} \)
73 \( 1 - 3.32e18T + 1.84e37T^{2} \)
79 \( 1 + 1.22e19iT - 8.96e37T^{2} \)
83 \( 1 - 2.23e19T + 2.40e38T^{2} \)
89 \( 1 + 3.15e19T + 9.72e38T^{2} \)
97 \( 1 - 8.67e19T + 5.43e39T^{2} \)
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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

−12.76049792118830256144059632827, −11.78016610218880053323857328551, −9.329284813358604840242772581562, −9.102923905340379529381103191895, −8.205796725586123100607262761447, −6.56044875152544139866064717476, −4.70709082597351467855981445129, −3.60849651178549515498535408379, −2.16245697359718579197996213742, −1.44161474658503928063586188892, 0.871011141413865222808233250509, 2.31038890964879586144933525988, 3.38615961108386387336805905649, 4.11416000771251319438698562819, 6.65038103472242188778423007994, 7.51394850822558280324840648704, 8.611424693508243213106895782918, 9.968059868690914821964076607886, 10.78429996279125513762035895801, 12.80997908334721484523521016779

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