↑

Properties

Label 2-108-9.7-c3-0-0
Degree $2$
Conductor $108$
Sign $-0.839 - 0.543i$
Analytic cond. $6.37220$
Root an. cond. $2.52432$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−6.92 − 11.9i)5-s + (−15.3 + 26.5i)7-s + (−21.9 + 38.0i)11-s + (6.11 + 10.5i)13-s − 76.0·17-s − 44.1·19-s + (−39.3 − 68.0i)23-s + (−33.3 + 57.7i)25-s + (−46.3 + 80.3i)29-s + (71.5 + 123. i)31-s + 425.·35-s − 32.4·37-s + (−167. − 290. i)41-s + (249. − 431. i)43-s + (−140. + 244. i)47-s + ⋯
L(s)  = 1  + (−0.619 − 1.07i)5-s + (−0.829 + 1.43i)7-s + (−0.601 + 1.04i)11-s + (0.130 + 0.226i)13-s − 1.08·17-s − 0.533·19-s + (−0.356 − 0.617i)23-s + (−0.266 + 0.461i)25-s + (−0.297 + 0.514i)29-s + (0.414 + 0.717i)31-s + 2.05·35-s − 0.144·37-s + (−0.639 − 1.10i)41-s + (0.883 − 1.53i)43-s + (−0.437 + 0.757i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 108 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(108\)    =    \(2^{2} \cdot 3^{3}\)
Sign: $-0.839 - 0.543i$
Analytic conductor: \(6.37220\)
Root analytic conductor: \(2.52432\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{108} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 108,\ (\ :3/2),\ -0.839 - 0.543i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.101381 + 0.342747i\)
\(L(\frac12)\) \(\approx\) \(0.101381 + 0.342747i\)
\(L(\frac{5}{2})\) 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 \( 1 + (6.92 + 11.9i)T + (-62.5 + 108. i)T^{2} \)
7 \( 1 + (15.3 - 26.5i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (21.9 - 38.0i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 + (-6.11 - 10.5i)T + (-1.09e3 + 1.90e3i)T^{2} \)
17 \( 1 + 76.0T + 4.91e3T^{2} \)
19 \( 1 + 44.1T + 6.85e3T^{2} \)
23 \( 1 + (39.3 + 68.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (46.3 - 80.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + (-71.5 - 123. i)T + (-1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 32.4T + 5.06e4T^{2} \)
41 \( 1 + (167. + 290. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-249. + 431. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 + (140. - 244. i)T + (-5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 628.T + 1.48e5T^{2} \)
59 \( 1 + (-252. - 437. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (185. - 322. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-81.3 - 140. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 433.T + 3.57e5T^{2} \)
73 \( 1 + 629.T + 3.89e5T^{2} \)
79 \( 1 + (86.3 - 149. i)T + (-2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 + (87.4 - 151. i)T + (-2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 336.T + 7.04e5T^{2} \)
97 \( 1 + (42.1 - 73.0i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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

−13.29670154789053497155666878545, −12.43635733177369012877843130152, −12.01416624165151729026167647548, −10.40518937939229141855552676753, −9.042256222058445192317126145184, −8.550069334183797666421594853323, −6.94385304599567988726099320141, −5.50951495613372034614665174932, −4.31112586165925539767884154710, −2.34826370692623216874775066081, 0.18563660148585931195432112021, 3.04706328400482937573617197524, 4.10051637422683345070103402478, 6.20517444479461668825884209700, 7.14747505049895992718645475207, 8.169109422463150787506551977660, 9.853130978317407017011754922790, 10.78991264197072477671999815237, 11.38700439896862608137036464759, 13.15281951288851388844291446793

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