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Properties

Label 2-112-112.53-c1-0-7
Degree $2$
Conductor $112$
Sign $0.262 - 0.964i$
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 + i)2-s + (0.5 + 1.86i)3-s + 2i·4-s + (0.866 − 3.23i)5-s + (−1.36 + 2.36i)6-s + (−1.73 − 2i)7-s + (−2 + 2i)8-s + (−0.633 + 0.366i)9-s + (4.09 − 2.36i)10-s + (−4.23 + 1.13i)11-s + (−3.73 + i)12-s + (0.267 − 0.267i)13-s + (0.267 − 3.73i)14-s + 6.46·15-s − 4·16-s + (0.232 − 0.401i)17-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + (0.288 + 1.07i)3-s + i·4-s + (0.387 − 1.44i)5-s + (−0.557 + 0.965i)6-s + (−0.654 − 0.755i)7-s + (−0.707 + 0.707i)8-s + (−0.211 + 0.122i)9-s + (1.29 − 0.748i)10-s + (−1.27 + 0.341i)11-s + (−1.07 + 0.288i)12-s + (0.0743 − 0.0743i)13-s + (0.0716 − 0.997i)14-s + 1.66·15-s − 16-s + (0.0562 − 0.0974i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.262 - 0.964i$
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: \((2,\ 112,\ (\ :1/2),\ 0.262 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19026 + 0.909408i\)
\(L(\frac12)\) \(\approx\) \(1.19026 + 0.909408i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1 - i)T \)
7 \( 1 + (1.73 + 2i)T \)
good3 \( 1 + (-0.5 - 1.86i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (-0.866 + 3.23i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (4.23 - 1.13i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.267 + 0.267i)T - 13iT^{2} \)
17 \( 1 + (-0.232 + 0.401i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.23 - 1.13i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.13 + 1.23i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.73 - 3.73i)T - 29iT^{2} \)
31 \( 1 + (0.133 - 0.232i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.86 - 10.6i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + 8.92iT - 41T^{2} \)
43 \( 1 + (0.464 + 0.464i)T + 43iT^{2} \)
47 \( 1 + (3.86 + 6.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.0 + 2.96i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (9.96 - 2.66i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.133 - 0.0358i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-1.96 - 7.33i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.46iT - 71T^{2} \)
73 \( 1 + (-2.76 - 1.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.330 + 0.571i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.46 + 8.46i)T - 83iT^{2} \)
89 \( 1 + (4.5 - 2.59i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−13.68886459055929666880713272744, −13.10818854777947759283931862692, −12.21312145236334582773812244187, −10.43898169690887556508257203536, −9.481796011209870622456345320148, −8.497196582956406258542520261480, −7.19444575880655847641669455476, −5.41386314944077732733944861136, −4.70514491732352094662851900158, −3.42177136837605976762229564734, 2.32140870906120728004430199755, 3.12933252734082075382580597383, 5.56632499461731680468894996119, 6.53939928113158581734795868057, 7.61626870263990658289046432823, 9.443874778242363644810683293079, 10.49779333312649755707594395470, 11.43792132306193209555213176695, 12.63915562174964162475426508051, 13.34533097650639679021201982507

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