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Properties

Label 2-112-112.93-c1-0-2
Degree $2$
Conductor $112$
Sign $0.704 - 0.710i$
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  + (−1.36 − 0.366i)2-s + (−0.133 + 0.5i)3-s + (1.73 + i)4-s + (0.232 + 0.866i)5-s + (0.366 − 0.633i)6-s + (−1.73 + 2i)7-s + (−1.99 − 2i)8-s + (2.36 + 1.36i)9-s − 1.26i·10-s + (2.86 + 0.767i)11-s + (−0.732 + 0.732i)12-s + (3.73 + 3.73i)13-s + (3.09 − 2.09i)14-s − 0.464·15-s + (1.99 + 3.46i)16-s + (−3.23 − 5.59i)17-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (−0.0773 + 0.288i)3-s + (0.866 + 0.5i)4-s + (0.103 + 0.387i)5-s + (0.149 − 0.258i)6-s + (−0.654 + 0.755i)7-s + (−0.707 − 0.707i)8-s + (0.788 + 0.455i)9-s − 0.400i·10-s + (0.864 + 0.231i)11-s + (−0.211 + 0.211i)12-s + (1.03 + 1.03i)13-s + (0.827 − 0.560i)14-s − 0.119·15-s + (0.499 + 0.866i)16-s + (−0.783 − 1.35i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 - 0.710i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 - 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.704 - 0.710i$
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: \((2,\ 112,\ (\ :1/2),\ 0.704 - 0.710i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.644793 + 0.268650i\)
\(L(\frac12)\) \(\approx\) \(0.644793 + 0.268650i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.366i)T \)
7 \( 1 + (1.73 - 2i)T \)
good3 \( 1 + (0.133 - 0.5i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (-0.232 - 0.866i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.86 - 0.767i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.73 - 3.73i)T + 13iT^{2} \)
17 \( 1 + (3.23 + 5.59i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.86 - 0.767i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.86 + 2.23i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.267 + 0.267i)T + 29iT^{2} \)
31 \( 1 + (1.86 + 3.23i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.303 - 1.13i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.92iT - 41T^{2} \)
43 \( 1 + (-6.46 + 6.46i)T - 43iT^{2} \)
47 \( 1 + (2.13 - 3.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.96 - 1.06i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-11.3 - 3.03i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (6.96 - 1.86i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-1.33 + 4.96i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.535iT - 71T^{2} \)
73 \( 1 + (6.23 - 3.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.33 + 14.4i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.53 - 1.53i)T + 83iT^{2} \)
89 \( 1 + (-4.5 - 2.59i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.92T + 97T^{2} \)
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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

−13.68035867043160337069524379457, −12.42387755297471617435466405357, −11.48678702618403913757107207015, −10.50171865875511417460398320615, −9.422755234396525407155551535769, −8.764729687717167569777778584744, −7.08643473120626839640111432384, −6.28430272947960894925521697447, −4.05715735782201210510796503964, −2.22385599022126582951094311583, 1.24146913354143854554639422157, 3.80884539421079449397900098444, 6.07263385205518676968056854941, 6.80795493234387284627500969275, 8.148913898394146148494356787245, 9.157109126001502427554888212124, 10.25321140762282906327605635047, 11.08679796110313081452107242966, 12.56809662573489871882903957140, 13.28340306065856332196711054303

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