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Properties

Label 2-112-112.53-c1-0-1
Degree $2$
Conductor $112$
Sign $0.917 - 0.398i$
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  + (−1.32 − 0.492i)2-s + (−0.221 − 0.827i)3-s + (1.51 + 1.30i)4-s + (−1.10 + 4.10i)5-s + (−0.113 + 1.20i)6-s + (2.50 + 0.856i)7-s + (−1.36 − 2.47i)8-s + (1.96 − 1.13i)9-s + (3.48 − 4.90i)10-s + (1.18 − 0.318i)11-s + (0.744 − 1.54i)12-s + (−1.73 + 1.73i)13-s + (−2.89 − 2.36i)14-s + 3.64·15-s + (0.592 + 3.95i)16-s + (−0.931 + 1.61i)17-s + ⋯
L(s)  = 1  + (−0.937 − 0.348i)2-s + (−0.128 − 0.477i)3-s + (0.757 + 0.652i)4-s + (−0.492 + 1.83i)5-s + (−0.0463 + 0.492i)6-s + (0.946 + 0.323i)7-s + (−0.483 − 0.875i)8-s + (0.653 − 0.377i)9-s + (1.10 − 1.55i)10-s + (0.358 − 0.0960i)11-s + (0.214 − 0.445i)12-s + (−0.480 + 0.480i)13-s + (−0.774 − 0.632i)14-s + 0.941·15-s + (0.148 + 0.988i)16-s + (−0.226 + 0.391i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 - 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.917 - 0.398i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.917 - 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685680 + 0.142603i\)
\(L(\frac12)\) \(\approx\) \(0.685680 + 0.142603i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 + 0.492i)T \)
7 \( 1 + (-2.50 - 0.856i)T \)
good3 \( 1 + (0.221 + 0.827i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (1.10 - 4.10i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.18 + 0.318i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (0.931 - 1.61i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.69 - 0.989i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (1.23 - 0.711i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.181 - 0.181i)T - 29iT^{2} \)
31 \( 1 + (-3.23 + 5.59i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.0637 - 0.237i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.440iT - 41T^{2} \)
43 \( 1 + (5.54 + 5.54i)T + 43iT^{2} \)
47 \( 1 + (3.61 + 6.26i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.69 + 2.59i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.63 + 0.438i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.01 + 2.41i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (2.57 + 9.59i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 11.2iT - 71T^{2} \)
73 \( 1 + (-8.13 - 4.69i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-6.52 - 11.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.06 - 3.06i)T - 83iT^{2} \)
89 \( 1 + (-5.66 + 3.26i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 1.70T + 97T^{2} \)
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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

−13.77220753967702649914868134282, −12.01344352499999394182495381858, −11.62765230096685948923754557606, −10.61234057499956872467452697940, −9.636112920152173904082703131221, −8.037709326283821441777962469765, −7.23475065392365505120329375599, −6.39597367574672614047290515645, −3.73505282491083373535246405740, −2.11498926547055012845580496282, 1.27607795198056493251944993304, 4.55174362047144137945473626020, 5.27975628552070929249913712668, 7.38036473495405439084415769686, 8.209736476126417396179534048946, 9.181298100896475809357145910132, 10.14403314433096038376133637445, 11.41055210596803391738437003375, 12.24867941633385644198346787991, 13.58433590867370070885461364528

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