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Properties

Label 2-112-112.93-c1-0-11
Degree $2$
Conductor $112$
Sign $-0.191 + 0.981i$
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  + (0.438 − 1.34i)2-s + (0.0145 − 0.0543i)3-s + (−1.61 − 1.17i)4-s + (−0.337 − 1.25i)5-s + (−0.0667 − 0.0434i)6-s + (−0.230 − 2.63i)7-s + (−2.29 + 1.65i)8-s + (2.59 + 1.49i)9-s + (−1.83 − 0.0979i)10-s + (1.50 + 0.402i)11-s + (−0.0876 + 0.0707i)12-s + (1.59 + 1.59i)13-s + (−3.64 − 0.844i)14-s − 0.0733·15-s + (1.22 + 3.80i)16-s + (1.46 + 2.54i)17-s + ⋯
L(s)  = 1  + (0.309 − 0.950i)2-s + (0.00841 − 0.0313i)3-s + (−0.807 − 0.589i)4-s + (−0.150 − 0.562i)5-s + (−0.0272 − 0.0177i)6-s + (−0.0872 − 0.996i)7-s + (−0.810 + 0.585i)8-s + (0.865 + 0.499i)9-s + (−0.581 − 0.0309i)10-s + (0.453 + 0.121i)11-s + (−0.0252 + 0.0204i)12-s + (0.442 + 0.442i)13-s + (−0.974 − 0.225i)14-s − 0.0189·15-s + (0.305 + 0.952i)16-s + (0.356 + 0.617i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.191 + 0.981i$
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: \((2,\ 112,\ (\ :1/2),\ -0.191 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.713152 - 0.866136i\)
\(L(\frac12)\) \(\approx\) \(0.713152 - 0.866136i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.438 + 1.34i)T \)
7 \( 1 + (0.230 + 2.63i)T \)
good3 \( 1 + (-0.0145 + 0.0543i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (0.337 + 1.25i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.50 - 0.402i)T + (9.52 + 5.5i)T^{2} \)
13 \( 1 + (-1.59 - 1.59i)T + 13iT^{2} \)
17 \( 1 + (-1.46 - 2.54i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (7.65 - 2.05i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.91 - 2.26i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.06 - 2.06i)T + 29iT^{2} \)
31 \( 1 + (3.14 + 5.43i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.40 + 5.24i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 7.34iT - 41T^{2} \)
43 \( 1 + (1.99 - 1.99i)T - 43iT^{2} \)
47 \( 1 + (-0.979 + 1.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (11.2 + 3.00i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.96 + 0.793i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (9.72 - 2.60i)T + (52.8 - 30.5i)T^{2} \)
67 \( 1 + (-0.566 + 2.11i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.26iT - 71T^{2} \)
73 \( 1 + (-12.2 + 7.06i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (0.961 - 1.66i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.82 - 8.82i)T + 83iT^{2} \)
89 \( 1 + (11.5 + 6.66i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 9.69T + 97T^{2} \)
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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

−13.00852011500009037679789251690, −12.59758731565536267269200057385, −11.12278933592713611048466233957, −10.42130124366076393762883243914, −9.300859452131144779336936831195, −8.072683102290370283971644006526, −6.47750063045161871999209502843, −4.68310675240542127629808848027, −3.86116028183780260387213853448, −1.52998524714208345601388138140, 3.23854653063871578418669468303, 4.77573700368747785553571012155, 6.25426198160868965512284696122, 7.01535999466512824219242490516, 8.490146638841496085553268485766, 9.320162411621610468088839044349, 10.76881960362073760696658812042, 12.24398891823818179505453371511, 12.88030274155149596662079534685, 14.16414742022540355013367227725

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