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Properties

Label 2-112-112.27-c1-0-7
Degree $2$
Conductor $112$
Sign $0.761 - 0.648i$
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 + (−2.89 + 2.37i)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 + (−0.772 + 0.634i)14-s − 0.902i·15-s + (0.854 − 0.520i)16-s − 0.275i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.761 - 0.648i$
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: \((2,\ 112,\ (\ :1/2),\ 0.761 - 0.648i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19982 + 0.441650i\)
\(L(\frac12)\) \(\approx\) \(1.19982 + 0.441650i\)
\(L(\frac{3}{2})\) 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

−13.81246047869051318495474188567, −12.81718812429454538423305708747, −12.39744962392156035541158704409, −10.19281517294956312431741495196, −9.113494346181470407419560814644, −8.113479908437551672128659821721, −7.39534700767378007369326854569, −5.59898649571865667129182042207, −4.96465726021758334280326831368, −2.37823808047272748234609508572, 2.30271973765317417995284274268, 3.70880413915913300254031349864, 4.98489051490523910599730273774, 6.80968634877713810443398681289, 8.494239811782008503042476742869, 9.619774650612803513057814645874, 10.31393023591043513592973966485, 11.17833587155533033338372778233, 12.43782884070037270301535886112, 13.84732674782955192399508274033

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