↑

Properties

Label 2-2e2-4.3-c30-0-2
Degree $2$
Conductor $4$
Sign $-0.853 - 0.520i$
Analytic cond. $22.8057$
Root an. cond. $4.77553$
Motivic weight $30$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−3.15e4 − 8.85e3i)2-s + 4.59e6i·3-s + (9.16e8 + 5.58e8i)4-s − 1.46e10·5-s + (4.07e10 − 1.45e11i)6-s + 9.18e12i·7-s + (−2.39e13 − 2.57e13i)8-s + 1.84e14·9-s + (4.60e14 + 1.29e14i)10-s − 4.03e15i·11-s + (−2.56e15 + 4.21e15i)12-s + 4.32e16·13-s + (8.13e16 − 2.89e17i)14-s − 6.71e16i·15-s + (5.28e17 + 1.02e18i)16-s − 6.32e17·17-s + ⋯
L(s)  = 1  + (−0.962 − 0.270i)2-s + 0.320i·3-s + (0.853 + 0.520i)4-s − 0.478·5-s + (0.0866 − 0.308i)6-s + 1.93i·7-s + (−0.681 − 0.731i)8-s + 0.897·9-s + (0.460 + 0.129i)10-s − 0.966i·11-s + (−0.166 + 0.273i)12-s + 0.845·13-s + (0.522 − 1.86i)14-s − 0.153i·15-s + (0.458 + 0.888i)16-s − 0.221·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(31-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+15) \, L(s)\cr =\mathstrut & (-0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4\)    =    \(2^{2}\)
Sign: $-0.853 - 0.520i$
Analytic conductor: \(22.8057\)
Root analytic conductor: \(4.77553\)
Motivic weight: \(30\)
Rational: no
Arithmetic: yes
Character: $\chi_{4} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4,\ (\ :15),\ -0.853 - 0.520i)\)

Particular Values

\(L(\frac{31}{2})\) \(\approx\) \(0.185689 + 0.661520i\)
\(L(\frac12)\) \(\approx\) \(0.185689 + 0.661520i\)
\(L(16)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.15e4 + 8.85e3i)T \)
good3 \( 1 - 4.59e6iT - 2.05e14T^{2} \)
5 \( 1 + 1.46e10T + 9.31e20T^{2} \)
7 \( 1 - 9.18e12iT - 2.25e25T^{2} \)
11 \( 1 + 4.03e15iT - 1.74e31T^{2} \)
13 \( 1 - 4.32e16T + 2.61e33T^{2} \)
17 \( 1 + 6.32e17T + 8.19e36T^{2} \)
19 \( 1 - 1.63e19iT - 2.30e38T^{2} \)
23 \( 1 + 1.15e19iT - 7.10e40T^{2} \)
29 \( 1 + 1.25e22T + 7.44e43T^{2} \)
31 \( 1 - 2.07e22iT - 5.50e44T^{2} \)
37 \( 1 - 1.65e23T + 1.11e47T^{2} \)
41 \( 1 + 7.85e23T + 2.41e48T^{2} \)
43 \( 1 + 1.58e24iT - 1.00e49T^{2} \)
47 \( 1 - 1.80e25iT - 1.45e50T^{2} \)
53 \( 1 + 4.74e25T + 5.34e51T^{2} \)
59 \( 1 + 2.91e26iT - 1.33e53T^{2} \)
61 \( 1 + 6.03e26T + 3.62e53T^{2} \)
67 \( 1 + 3.55e27iT - 6.05e54T^{2} \)
71 \( 1 - 5.42e27iT - 3.44e55T^{2} \)
73 \( 1 - 6.40e27T + 7.93e55T^{2} \)
79 \( 1 + 4.23e28iT - 8.48e56T^{2} \)
83 \( 1 - 4.11e28iT - 3.73e57T^{2} \)
89 \( 1 + 3.08e29T + 3.03e58T^{2} \)
97 \( 1 + 2.56e29T + 4.01e59T^{2} \)
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

−18.37630600202793741297903017366, −16.18053041085072636053076489697, −15.36825475758298983266900245460, −12.46470508146598261381564897803, −11.16125649689710704570471884906, −9.336414176637045195311510094142, −8.156623218311152861905026877634, −5.98778945532032183321823032243, −3.41687732261880330921837622351, −1.70135151736718443727783498521, 0.33555476370600057654406371424, 1.55002605541363024802057154338, 4.12195059186766760120943468782, 6.89105637681032075195969004862, 7.67283804475280915210814468639, 9.830250450887259788372105230524, 11.14167635189513464812319531702, 13.34100588406416990942670592668, 15.36562308148332963775714557830, 16.80631792598973169171859942102

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