↑

Properties

Label 2-48e2-16.5-c1-0-37
Degree $2$
Conductor $2304$
Sign $-0.991 - 0.130i$
Analytic cond. $18.3975$
Root an. cond. $4.28923$
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  + (−2.73 − 2.73i)5-s − 2.44i·7-s + (−1.03 − 1.03i)11-s + (3 − 3i)13-s − 3.46·17-s + (5.27 − 5.27i)19-s − 2.82i·23-s + 9.92i·25-s + (6.19 − 6.19i)29-s − 7.34·31-s + (−6.69 + 6.69i)35-s + (0.464 + 0.464i)37-s + 4.53i·41-s + (2.44 + 2.44i)43-s + 2.82·47-s + ⋯
L(s)  = 1  + (−1.22 − 1.22i)5-s − 0.925i·7-s + (−0.312 − 0.312i)11-s + (0.832 − 0.832i)13-s − 0.840·17-s + (1.21 − 1.21i)19-s − 0.589i·23-s + 1.98i·25-s + (1.15 − 1.15i)29-s − 1.31·31-s + (−1.13 + 1.13i)35-s + (0.0762 + 0.0762i)37-s + 0.708i·41-s + (0.373 + 0.373i)43-s + 0.412·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2304\)    =    \(2^{8} \cdot 3^{2}\)
Sign: $-0.991 - 0.130i$
Analytic conductor: \(18.3975\)
Root analytic conductor: \(4.28923\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2304} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2304,\ (\ :1/2),\ -0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9265778165\)
\(L(\frac12)\) \(\approx\) \(0.9265778165\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good5 \( 1 + (2.73 + 2.73i)T + 5iT^{2} \)
7 \( 1 + 2.44iT - 7T^{2} \)
11 \( 1 + (1.03 + 1.03i)T + 11iT^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + 3.46T + 17T^{2} \)
19 \( 1 + (-5.27 + 5.27i)T - 19iT^{2} \)
23 \( 1 + 2.82iT - 23T^{2} \)
29 \( 1 + (-6.19 + 6.19i)T - 29iT^{2} \)
31 \( 1 + 7.34T + 31T^{2} \)
37 \( 1 + (-0.464 - 0.464i)T + 37iT^{2} \)
41 \( 1 - 4.53iT - 41T^{2} \)
43 \( 1 + (-2.44 - 2.44i)T + 43iT^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + (3.26 + 3.26i)T + 53iT^{2} \)
59 \( 1 + (9.79 + 9.79i)T + 59iT^{2} \)
61 \( 1 + (-0.464 + 0.464i)T - 61iT^{2} \)
67 \( 1 + (10.5 - 10.5i)T - 67iT^{2} \)
71 \( 1 - 6.41iT - 71T^{2} \)
73 \( 1 + 4iT - 73T^{2} \)
79 \( 1 + 2.44T + 79T^{2} \)
83 \( 1 + (3.86 - 3.86i)T - 83iT^{2} \)
89 \( 1 - 4.92iT - 89T^{2} \)
97 \( 1 - 1.07T + 97T^{2} \)
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

−8.555661507232098346492347426438, −7.84670606886784775517883183672, −7.33811148429329967867489364382, −6.29101064806501705700687173913, −5.19370550391601629422596982385, −4.53437189872939734550334756276, −3.84328661198962257042717748325, −2.90189257038087699060938300773, −1.08731063889929318421155516574, −0.37768616686863925024896562651, 1.74192111962352485336687987767, 2.92540299877419486611044125122, 3.58360420534588175048098224362, 4.44927346785620056647525835312, 5.60758033552344655661830513459, 6.36056558366970873994280369803, 7.26687415714563300341461013409, 7.65004295227604530522864527720, 8.706784097428380851767244930568, 9.183307838433925396952850099453

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