↑

Properties

Label 2-304-304.227-c1-0-3
Degree $2$
Conductor $304$
Sign $-0.909 + 0.415i$
Analytic cond. $2.42745$
Root an. cond. $1.55802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.18 + 0.779i)2-s + (1.07 + 1.07i)3-s + (0.785 − 1.83i)4-s + (−1.37 + 1.37i)5-s + (−2.11 − 0.431i)6-s − 4.00·7-s + (0.506 + 2.78i)8-s − 0.676i·9-s + (0.550 − 2.69i)10-s + (−2.44 − 2.44i)11-s + (2.82 − 1.13i)12-s + (−3.86 + 3.86i)13-s + (4.72 − 3.11i)14-s − 2.96·15-s + (−2.76 − 2.88i)16-s + 0.770·17-s + ⋯
L(s)  = 1  + (−0.834 + 0.551i)2-s + (0.622 + 0.622i)3-s + (0.392 − 0.919i)4-s + (−0.614 + 0.614i)5-s + (−0.862 − 0.176i)6-s − 1.51·7-s + (0.179 + 0.983i)8-s − 0.225i·9-s + (0.174 − 0.851i)10-s + (−0.735 − 0.735i)11-s + (0.816 − 0.327i)12-s + (−1.07 + 1.07i)13-s + (1.26 − 0.833i)14-s − 0.764·15-s + (−0.691 − 0.722i)16-s + 0.186·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 + 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(304\)    =    \(2^{4} \cdot 19\)
Sign: $-0.909 + 0.415i$
Analytic conductor: \(2.42745\)
Root analytic conductor: \(1.55802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{304} (227, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 304,\ (\ :1/2),\ -0.909 + 0.415i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0609601 - 0.280448i\)
\(L(\frac12)\) \(\approx\) \(0.0609601 - 0.280448i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.18 - 0.779i)T \)
19 \( 1 + (2.90 - 3.25i)T \)
good3 \( 1 + (-1.07 - 1.07i)T + 3iT^{2} \)
5 \( 1 + (1.37 - 1.37i)T - 5iT^{2} \)
7 \( 1 + 4.00T + 7T^{2} \)
11 \( 1 + (2.44 + 2.44i)T + 11iT^{2} \)
13 \( 1 + (3.86 - 3.86i)T - 13iT^{2} \)
17 \( 1 - 0.770T + 17T^{2} \)
23 \( 1 - 7.83T + 23T^{2} \)
29 \( 1 + (4.22 - 4.22i)T - 29iT^{2} \)
31 \( 1 + 6.35T + 31T^{2} \)
37 \( 1 + (1.65 + 1.65i)T + 37iT^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 + (7.25 + 7.25i)T + 43iT^{2} \)
47 \( 1 + 3.06iT - 47T^{2} \)
53 \( 1 + (-8.94 - 8.94i)T + 53iT^{2} \)
59 \( 1 + (8.06 - 8.06i)T - 59iT^{2} \)
61 \( 1 + (-5.67 - 5.67i)T + 61iT^{2} \)
67 \( 1 + (-6.75 - 6.75i)T + 67iT^{2} \)
71 \( 1 + 5.29iT - 71T^{2} \)
73 \( 1 - 7.64iT - 73T^{2} \)
79 \( 1 + 4.93T + 79T^{2} \)
83 \( 1 + (-1.44 + 1.44i)T - 83iT^{2} \)
89 \( 1 + 0.971T + 89T^{2} \)
97 \( 1 - 6.62iT - 97T^{2} \)
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

−12.09519696905975066569011502437, −10.89544640873081169235113761048, −10.17921206851389636853003028598, −9.272620674306761331664934132075, −8.761467839690188149896498886359, −7.31736193770648618561013583165, −6.80267162632408668496991671350, −5.50799460903944719559130237672, −3.76979626805814265852092971448, −2.76010437377592022242125982880, 0.23459097572602325022571272489, 2.38013454723545765398474515189, 3.27440429073150842973065477298, 4.92349333842764815896061150516, 6.83701239826336474959936075096, 7.54485053026458265637853551061, 8.311910186809209273252306250350, 9.341120032596718271236366239903, 10.06350016996389663380109977865, 11.07528480364983314015969894430

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