↑

Properties

Label 2-2e2-4.3-c22-0-2
Degree $2$
Conductor $4$
Sign $-0.520 - 0.853i$
Analytic cond. $12.2682$
Root an. cond. $3.50261$
Motivic weight $22$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.78e3 + 1.00e3i)2-s − 1.27e5i·3-s + (2.18e6 + 3.58e6i)4-s − 6.52e7·5-s + (1.28e8 − 2.28e8i)6-s + 3.27e9i·7-s + (3.12e8 + 8.58e9i)8-s + 1.50e10·9-s + (−1.16e11 − 6.53e10i)10-s + 2.30e11i·11-s + (4.57e11 − 2.79e11i)12-s − 1.19e12·13-s + (−3.28e12 + 5.85e12i)14-s + 8.34e12i·15-s + (−8.04e12 + 1.56e13i)16-s − 2.07e13·17-s + ⋯
L(s)  = 1  + (0.872 + 0.489i)2-s − 0.721i·3-s + (0.520 + 0.853i)4-s − 1.33·5-s + (0.353 − 0.629i)6-s + 1.65i·7-s + (0.0363 + 0.999i)8-s + 0.479·9-s + (−1.16 − 0.653i)10-s + 0.808i·11-s + (0.616 − 0.375i)12-s − 0.668·13-s + (−0.811 + 1.44i)14-s + 0.964i·15-s + (−0.457 + 0.889i)16-s − 0.605·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(23-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+11) \, L(s)\cr =\mathstrut & (-0.520 - 0.853i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.520 - 0.853i$
Analytic conductor: \(12.2682\)
Root analytic conductor: \(3.50261\)
Motivic weight: \(22\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :11),\ -0.520 - 0.853i)\)

Particular Values

\(L(\frac{23}{2})\) \(\approx\) \(0.928890 + 1.65490i\)
\(L(\frac12)\) \(\approx\) \(0.928890 + 1.65490i\)
\(L(12)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.78e3 - 1.00e3i)T \)
good3 \( 1 + 1.27e5iT - 3.13e10T^{2} \)
5 \( 1 + 6.52e7T + 2.38e15T^{2} \)
7 \( 1 - 3.27e9iT - 3.90e18T^{2} \)
11 \( 1 - 2.30e11iT - 8.14e22T^{2} \)
13 \( 1 + 1.19e12T + 3.21e24T^{2} \)
17 \( 1 + 2.07e13T + 1.17e27T^{2} \)
19 \( 1 - 6.48e12iT - 1.35e28T^{2} \)
23 \( 1 + 1.02e15iT - 9.07e29T^{2} \)
29 \( 1 - 1.56e16T + 1.48e32T^{2} \)
31 \( 1 - 2.85e16iT - 6.45e32T^{2} \)
37 \( 1 - 2.10e17T + 3.16e34T^{2} \)
41 \( 1 + 5.57e17T + 3.02e35T^{2} \)
43 \( 1 - 8.07e17iT - 8.63e35T^{2} \)
47 \( 1 - 8.59e17iT - 6.11e36T^{2} \)
53 \( 1 - 1.56e19T + 8.59e37T^{2} \)
59 \( 1 + 7.82e18iT - 9.09e38T^{2} \)
61 \( 1 - 1.24e19T + 1.89e39T^{2} \)
67 \( 1 + 1.36e20iT - 1.49e40T^{2} \)
71 \( 1 - 2.01e20iT - 5.34e40T^{2} \)
73 \( 1 - 2.57e19T + 9.84e40T^{2} \)
79 \( 1 - 8.27e20iT - 5.59e41T^{2} \)
83 \( 1 - 6.46e20iT - 1.65e42T^{2} \)
89 \( 1 - 1.96e21T + 7.70e42T^{2} \)
97 \( 1 + 3.08e21T + 5.11e43T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.70737566837619304117040155646, −18.17341210991862756578689197969, −15.85744871135991380698330700351, −14.93443792530580567474419444369, −12.57707325189159633857224729399, −11.94016088019737366437126963462, −8.234746449250496800075263976937, −6.77529425572115925125558215218, −4.63479339478341816075335195608, −2.46892955636887393409899343037, 0.62794904754211009893781078953, 3.62663048974497898784021607055, 4.47193448457790643944467216575, 7.26961728579851245555767867257, 10.25206245928393168985267982738, 11.49571808959048067249895985113, 13.48460747172545426130838904468, 15.21077463865097310419331269112, 16.41681514117344189699228477098, 19.37947572380019122746524770082

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