↑

Properties

Label 2-112-112.109-c1-0-5
Degree $2$
Conductor $112$
Sign $0.474 - 0.880i$
Analytic cond. $0.894324$
Root an. cond. $0.945687$
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  + (0.945 + 1.05i)2-s + (−0.0543 + 0.0145i)3-s + (−0.212 + 1.98i)4-s + (1.25 + 0.337i)5-s + (−0.0667 − 0.0434i)6-s + (0.230 − 2.63i)7-s + (−2.29 + 1.65i)8-s + (−2.59 + 1.49i)9-s + (0.834 + 1.64i)10-s + (−0.402 − 1.50i)11-s + (−0.0174 − 0.111i)12-s + (1.59 + 1.59i)13-s + (2.99 − 2.24i)14-s − 0.0733·15-s + (−3.90 − 0.845i)16-s + (1.46 − 2.54i)17-s + ⋯
L(s)  = 1  + (0.668 + 0.743i)2-s + (−0.0313 + 0.00841i)3-s + (−0.106 + 0.994i)4-s + (0.562 + 0.150i)5-s + (−0.0272 − 0.0177i)6-s + (0.0872 − 0.996i)7-s + (−0.810 + 0.585i)8-s + (−0.865 + 0.499i)9-s + (0.263 + 0.519i)10-s + (−0.121 − 0.453i)11-s + (−0.00502 − 0.0321i)12-s + (0.442 + 0.442i)13-s + (0.799 − 0.601i)14-s − 0.0189·15-s + (−0.977 − 0.211i)16-s + (0.356 − 0.617i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 - 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.474 - 0.880i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.474 - 0.880i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21201 + 0.723446i\)
\(L(\frac12)\) \(\approx\) \(1.21201 + 0.723446i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.945 - 1.05i)T \)
7 \( 1 + (-0.230 + 2.63i)T \)
good3 \( 1 + (0.0543 - 0.0145i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (-1.25 - 0.337i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.402 + 1.50i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.59 - 1.59i)T + 13iT^{2} \)
17 \( 1 + (-1.46 + 2.54i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.05 + 7.65i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.91 - 2.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 2.06i)T + 29iT^{2} \)
31 \( 1 + (3.14 - 5.43i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.24 - 1.40i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + (1.99 - 1.99i)T - 43iT^{2} \)
47 \( 1 + (-0.979 - 1.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.00 - 11.2i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.793 - 2.96i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.60 + 9.72i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (2.11 - 0.566i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (12.2 + 7.06i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.961 + 1.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.82 - 8.82i)T + 83iT^{2} \)
89 \( 1 + (-11.5 + 6.66i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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

−13.75222906351434902105626068028, −13.38512119528370994043284587791, −11.74468186452325975826215428078, −10.90691309229012596179151776711, −9.372435281483526536692655693436, −8.131235925897816416629690965588, −7.02367579546490354945914334640, −5.86579363568618408771433371894, −4.66283389861312256193742803268, −3.01375429849495455385234457604, 2.10043419376170000453392442310, 3.69519762220999174985568987320, 5.60875369465887346292406783131, 5.98173836708463501571023577683, 8.250365780401474909551627317234, 9.456394306291641582059882117448, 10.33533990419680629958955302896, 11.71072578972852642299122721982, 12.29349815368305946019019251542, 13.31542402340325493475307531425

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