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Properties

Label 2-112-112.93-c1-0-5
Degree $2$
Conductor $112$
Sign $0.936 - 0.351i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.07 − 0.922i)2-s + (−0.781 + 2.91i)3-s + (0.299 − 1.97i)4-s + (0.199 + 0.745i)5-s + (1.85 + 3.84i)6-s + (2.51 + 0.830i)7-s + (−1.50 − 2.39i)8-s + (−5.28 − 3.05i)9-s + (0.901 + 0.615i)10-s + (1.24 + 0.333i)11-s + (5.52 + 2.41i)12-s + (−0.919 − 0.919i)13-s + (3.45 − 1.42i)14-s − 2.32·15-s + (−3.82 − 1.18i)16-s + (−3.95 − 6.85i)17-s + ⋯
L(s)  = 1  + (0.758 − 0.651i)2-s + (−0.450 + 1.68i)3-s + (0.149 − 0.988i)4-s + (0.0893 + 0.333i)5-s + (0.755 + 1.57i)6-s + (0.949 + 0.313i)7-s + (−0.530 − 0.847i)8-s + (−1.76 − 1.01i)9-s + (0.285 + 0.194i)10-s + (0.374 + 0.100i)11-s + (1.59 + 0.698i)12-s + (−0.254 − 0.254i)13-s + (0.924 − 0.380i)14-s − 0.601·15-s + (−0.955 − 0.296i)16-s + (−0.959 − 1.66i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.936 - 0.351i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.936 - 0.351i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (93, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.936 - 0.351i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.33946 + 0.243271i\)
\(L(\frac12)\) \(\approx\) \(1.33946 + 0.243271i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.07 + 0.922i)T \)
7 \( 1 + (-2.51 - 0.830i)T \)
good3 \( 1 + (0.781 - 2.91i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.199 - 0.745i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.24 - 0.333i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.919 + 0.919i)T + 13iT^{2} \)
17 \( 1 + (3.95 + 6.85i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.78 - 0.478i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.33 + 1.92i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.25 - 5.25i)T + 29iT^{2} \)
31 \( 1 + (-2.44 - 4.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.343 - 1.28i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 2.84iT - 41T^{2} \)
43 \( 1 + (-0.585 + 0.585i)T - 43iT^{2} \)
47 \( 1 + (2.86 - 4.95i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.51 + 2.54i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (8.93 + 2.39i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-6.38 + 1.71i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.59 - 5.94i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.99iT - 71T^{2} \)
73 \( 1 + (-6.69 + 3.86i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (4.63 - 8.02i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.78 + 4.78i)T + 83iT^{2} \)
89 \( 1 + (-1.84 - 1.06i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.01T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.11069952921201501081905849492, −12.30580673244057846554884999279, −11.38904855643914680617351469213, −10.75096019504295181996038029804, −9.828405644697570819733771378288, −8.797869350418650458708526589130, −6.48667324886643795440104701158, −5.02122857654024970113831737920, −4.52592964441404018047617928301, −2.88562693052258627642202600162, 1.95083079280429223295931935750, 4.43302526919653184820021800439, 5.89731248846478733855673325169, 6.72856855913739169758062408332, 7.86176007981755718294485327920, 8.557350511497273713354859006413, 11.02457978899295667767001578006, 11.88097094906254906077550223170, 12.71997553657822419074582110188, 13.49635247461541295575567418703

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