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Properties

Label 2-112-112.27-c1-0-3
Degree $2$
Conductor $112$
Sign $-0.922 - 0.386i$
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  + (0.192 + 1.40i)2-s + (−1.03 + 1.03i)3-s + (−1.92 + 0.540i)4-s + (−1.68 + 1.68i)5-s + (−1.65 − 1.25i)6-s + (1.38 − 2.25i)7-s + (−1.12 − 2.59i)8-s + 0.851i·9-s + (−2.68 − 2.03i)10-s + (−2 + 2i)11-s + (1.43 − 2.55i)12-s + (4.80 + 4.80i)13-s + (3.42 + 1.50i)14-s − 3.49i·15-s + (3.41 − 2.08i)16-s + 1.13i·17-s + ⋯
L(s)  = 1  + (0.136 + 0.990i)2-s + (−0.598 + 0.598i)3-s + (−0.962 + 0.270i)4-s + (−0.754 + 0.754i)5-s + (−0.674 − 0.511i)6-s + (0.523 − 0.851i)7-s + (−0.398 − 0.917i)8-s + 0.283i·9-s + (−0.850 − 0.644i)10-s + (−0.603 + 0.603i)11-s + (0.414 − 0.737i)12-s + (1.33 + 1.33i)13-s + (0.915 + 0.402i)14-s − 0.902i·15-s + (0.854 − 0.520i)16-s + 0.275i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.922 - 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.922 - 0.386i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ -0.922 - 0.386i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.146752 + 0.729696i\)
\(L(\frac12)\) \(\approx\) \(0.146752 + 0.729696i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.192 - 1.40i)T \)
7 \( 1 + (-1.38 + 2.25i)T \)
good3 \( 1 + (1.03 - 1.03i)T - 3iT^{2} \)
5 \( 1 + (1.68 - 1.68i)T - 5iT^{2} \)
11 \( 1 + (2 - 2i)T - 11iT^{2} \)
13 \( 1 + (-4.80 - 4.80i)T + 13iT^{2} \)
17 \( 1 - 1.13iT - 17T^{2} \)
19 \( 1 + (-1.21 + 1.21i)T - 19iT^{2} \)
23 \( 1 + 1.33T + 23T^{2} \)
29 \( 1 + (-5.26 + 5.26i)T - 29iT^{2} \)
31 \( 1 - 8.31T + 31T^{2} \)
37 \( 1 + (4.18 + 4.18i)T + 37iT^{2} \)
41 \( 1 + 1.63T + 41T^{2} \)
43 \( 1 + (1.33 - 1.33i)T - 43iT^{2} \)
47 \( 1 + 1.93T + 47T^{2} \)
53 \( 1 + (-6.34 - 6.34i)T + 53iT^{2} \)
59 \( 1 + (3.29 + 3.29i)T + 59iT^{2} \)
61 \( 1 + (2.04 + 2.04i)T + 61iT^{2} \)
67 \( 1 + (-0.107 - 0.107i)T + 67iT^{2} \)
71 \( 1 - 13.0T + 71T^{2} \)
73 \( 1 + 6.24T + 73T^{2} \)
79 \( 1 + 4.51iT - 79T^{2} \)
83 \( 1 + (9.71 - 9.71i)T - 83iT^{2} \)
89 \( 1 - 11.6T + 89T^{2} \)
97 \( 1 + 3.23iT - 97T^{2} \)
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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

−14.10572824020196919819208487086, −13.51540814058755438979682375222, −11.81827915467312053844047100489, −10.91166541185598078528150815423, −9.982993003406729962238897660024, −8.354313376755484064203154516895, −7.39742686368951670438440055432, −6.34290047298647589593017292692, −4.75514813811223899333045193066, −3.93927389434580663705477360702, 0.957600385120806555636265257852, 3.26942352993869024638115788817, 5.00284256337927263150859333553, 5.97645680441566660340276172263, 8.187774533534802220933254791929, 8.640419433526620364083215101381, 10.34733222764572731921952174640, 11.45008798007149215065669699947, 12.07918082459727220854390396380, 12.79911071454663915645029074691

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