Properties

Label 2-112-7.2-c1-0-2
Degree $2$
Conductor $112$
Sign $0.701 + 0.712i$
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  + (−0.5 − 0.866i)3-s + (0.5 − 0.866i)5-s + (2 − 1.73i)7-s + (1 − 1.73i)9-s + (1.5 + 2.59i)11-s − 6·13-s − 0.999·15-s + (2.5 + 4.33i)17-s + (0.5 − 0.866i)19-s + (−2.5 − 0.866i)21-s + (−3.5 + 6.06i)23-s + (2 + 3.46i)25-s − 5·27-s + 2·29-s + (−2.5 − 4.33i)31-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (0.223 − 0.387i)5-s + (0.755 − 0.654i)7-s + (0.333 − 0.577i)9-s + (0.452 + 0.783i)11-s − 1.66·13-s − 0.258·15-s + (0.606 + 1.05i)17-s + (0.114 − 0.198i)19-s + (−0.545 − 0.188i)21-s + (−0.729 + 1.26i)23-s + (0.400 + 0.692i)25-s − 0.962·27-s + 0.371·29-s + (−0.449 − 0.777i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.701 + 0.712i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.701 + 0.712i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.701 + 0.712i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.953206 - 0.399472i\)
\(L(\frac12)\) \(\approx\) \(0.953206 - 0.399472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-2 + 1.73i)T \)
good3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 6T + 13T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.5 - 6.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.5 - 2.59i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-2.5 + 4.33i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.5 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2.5 + 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.5 + 7.79i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (3.5 + 6.06i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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

−13.36637103897624603813564426192, −12.36267530916875766472585257593, −11.73197044916876882571546769312, −10.23119771454001353389512992432, −9.376302927091470684983633768693, −7.74974602213413377460865354291, −7.00594909735142233711115444878, −5.44255037315216003872634593352, −4.11761957989625921924448635341, −1.60478291472805514387162796273, 2.52824303702105238294123305358, 4.59725453899486130670035939670, 5.57360598038331244081150557594, 7.14043922179287005866057068800, 8.379656881043203736225585759727, 9.698304938160205601973786242431, 10.57555685519585205276805160244, 11.65692809887798795583277751402, 12.49513974671558037969982310582, 14.26128449298071632015483819022

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