↑

Properties

Label 2-117-13.10-c3-0-9
Degree $2$
Conductor $117$
Sign $0.906 + 0.422i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
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  + (0.794 − 0.458i)2-s + (−3.57 + 6.19i)4-s − 15.4i·5-s + (17.8 + 10.2i)7-s + 13.9i·8-s + (−7.09 − 12.2i)10-s + (57.0 − 32.9i)11-s + (19.2 − 42.7i)13-s + 18.8·14-s + (−22.2 − 38.5i)16-s + (−22.1 + 38.3i)17-s + (127. + 73.5i)19-s + (95.9 + 55.3i)20-s + (30.2 − 52.3i)22-s + (−26.5 − 46.0i)23-s + ⋯
L(s)  = 1  + (0.280 − 0.162i)2-s + (−0.447 + 0.774i)4-s − 1.38i·5-s + (0.962 + 0.555i)7-s + 0.614i·8-s + (−0.224 − 0.388i)10-s + (1.56 − 0.902i)11-s + (0.410 − 0.911i)13-s + 0.360·14-s + (−0.347 − 0.602i)16-s + (−0.315 + 0.547i)17-s + (1.53 + 0.887i)19-s + (1.07 + 0.619i)20-s + (0.292 − 0.506i)22-s + (−0.240 − 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.906 + 0.422i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.906 + 0.422i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.89910 - 0.420560i\)
\(L(\frac12)\) \(\approx\) \(1.89910 - 0.420560i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-19.2 + 42.7i)T \)
good2 \( 1 + (-0.794 + 0.458i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 15.4iT - 125T^{2} \)
7 \( 1 + (-17.8 - 10.2i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-57.0 + 32.9i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (22.1 - 38.3i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-127. - 73.5i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (26.5 + 46.0i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (19.3 + 33.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 88.3iT - 2.97e4T^{2} \)
37 \( 1 + (-68.3 + 39.4i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (307. - 177. i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (203. - 353. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 67.9iT - 1.03e5T^{2} \)
53 \( 1 + 226.T + 1.48e5T^{2} \)
59 \( 1 + (-123. - 71.0i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (133. - 231. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-356. + 205. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-79.2 - 45.7i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 63.1iT - 3.89e5T^{2} \)
79 \( 1 + 287.T + 4.93e5T^{2} \)
83 \( 1 + 373. iT - 5.71e5T^{2} \)
89 \( 1 + (103. - 59.7i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-480. - 277. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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

−12.90686358104427131210197075173, −11.96509489497213399006040753947, −11.42921653479934102057874593895, −9.466091164858474910083942093318, −8.489075147010780920092029148894, −8.063685590066038915066592853400, −5.84613543060394813465187043530, −4.76838868846152978875898021096, −3.56618314866364362492282010876, −1.26245744090844475105604671636, 1.55615348120379106292653232142, 3.78767934683981654292035265483, 4.97478511427926818972355170378, 6.66499234857858236127867815930, 7.15360097545657682425968391316, 9.083443173977975232363146332417, 9.981652784309459153664440721937, 11.13385637267798234724210777387, 11.77609541734272372095468493940, 13.78095059889836858943815941766

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