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Properties

Label 2-112-16.5-c1-0-9
Degree $2$
Conductor $112$
Sign $0.999 - 0.0234i$
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  + (1.35 + 0.402i)2-s + (0.631 − 0.631i)3-s + (1.67 + 1.09i)4-s + (−2.34 − 2.34i)5-s + (1.10 − 0.601i)6-s + i·7-s + (1.83 + 2.15i)8-s + 2.20i·9-s + (−2.23 − 4.11i)10-s + (−2.18 − 2.18i)11-s + (1.74 − 0.368i)12-s + (−4.03 + 4.03i)13-s + (−0.402 + 1.35i)14-s − 2.95·15-s + (1.61 + 3.65i)16-s + 0.347·17-s + ⋯
L(s)  = 1  + (0.958 + 0.284i)2-s + (0.364 − 0.364i)3-s + (0.837 + 0.545i)4-s + (−1.04 − 1.04i)5-s + (0.453 − 0.245i)6-s + 0.377i·7-s + (0.647 + 0.761i)8-s + 0.734i·9-s + (−0.706 − 1.30i)10-s + (−0.658 − 0.658i)11-s + (0.504 − 0.106i)12-s + (−1.11 + 1.11i)13-s + (−0.107 + 0.362i)14-s − 0.763·15-s + (0.404 + 0.914i)16-s + 0.0843·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0234i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.999 - 0.0234i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.999 - 0.0234i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.63181 + 0.0191506i\)
\(L(\frac12)\) \(\approx\) \(1.63181 + 0.0191506i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.402i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-0.631 + 0.631i)T - 3iT^{2} \)
5 \( 1 + (2.34 + 2.34i)T + 5iT^{2} \)
11 \( 1 + (2.18 + 2.18i)T + 11iT^{2} \)
13 \( 1 + (4.03 - 4.03i)T - 13iT^{2} \)
17 \( 1 - 0.347T + 17T^{2} \)
19 \( 1 + (-4.26 + 4.26i)T - 19iT^{2} \)
23 \( 1 + 6.23iT - 23T^{2} \)
29 \( 1 + (-1.21 + 1.21i)T - 29iT^{2} \)
31 \( 1 + 1.26T + 31T^{2} \)
37 \( 1 + (6.42 + 6.42i)T + 37iT^{2} \)
41 \( 1 - 2.68iT - 41T^{2} \)
43 \( 1 + (-4.05 - 4.05i)T + 43iT^{2} \)
47 \( 1 - 4.64T + 47T^{2} \)
53 \( 1 + (-8.44 - 8.44i)T + 53iT^{2} \)
59 \( 1 + (5.17 + 5.17i)T + 59iT^{2} \)
61 \( 1 + (0.00533 - 0.00533i)T - 61iT^{2} \)
67 \( 1 + (-3.02 + 3.02i)T - 67iT^{2} \)
71 \( 1 + 0.828iT - 71T^{2} \)
73 \( 1 - 6.25iT - 73T^{2} \)
79 \( 1 + 0.755T + 79T^{2} \)
83 \( 1 + (3.66 - 3.66i)T - 83iT^{2} \)
89 \( 1 + 6.24iT - 89T^{2} \)
97 \( 1 - 2.18T + 97T^{2} \)
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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

−13.62143992723023950505939897784, −12.61184905687147728233341903259, −11.97344540977454616807203715544, −10.93959320310803709520834113866, −8.932530655441989424166245591233, −7.973216726068694192769810681305, −7.11611896802095498919253053791, −5.27882893478860935560608674008, −4.42266036673609360049994048723, −2.60121483000502965738846905038, 2.99089934094447321131846289281, 3.81310234451024353648200349186, 5.35205660603915262583319979219, 7.08096363867342351923233489093, 7.70842044675427740945734019863, 9.903273677944564254804769828635, 10.48737531092354765130818246929, 11.79290231885412298457157636187, 12.40936711067425047817763790705, 13.79815535045623441834304652905

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