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Properties

Label 2-112-28.3-c1-0-3
Degree $2$
Conductor $112$
Sign $0.605 + 0.795i$
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  + (1.32 − 2.29i)3-s + (−1.5 + 0.866i)5-s + 2.64·7-s + (−2 − 3.46i)9-s + (−3.96 − 2.29i)11-s + 3.46i·13-s + 4.58i·15-s + (4.5 + 2.59i)17-s + (1.32 + 2.29i)19-s + (3.50 − 6.06i)21-s + (−3.96 + 2.29i)23-s + (−1 + 1.73i)25-s − 2.64·27-s + (−1.32 + 2.29i)31-s + (−10.5 + 6.06i)33-s + ⋯
L(s)  = 1  + (0.763 − 1.32i)3-s + (−0.670 + 0.387i)5-s + 0.999·7-s + (−0.666 − 1.15i)9-s + (−1.19 − 0.690i)11-s + 0.960i·13-s + 1.18i·15-s + (1.09 + 0.630i)17-s + (0.303 + 0.525i)19-s + (0.763 − 1.32i)21-s + (−0.827 + 0.477i)23-s + (−0.200 + 0.346i)25-s − 0.509·27-s + (−0.237 + 0.411i)31-s + (−1.82 + 1.05i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08503 - 0.537879i\)
\(L(\frac12)\) \(\approx\) \(1.08503 - 0.537879i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - 2.64T \)
good3 \( 1 + (-1.32 + 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.96 + 2.29i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 3.46iT - 13T^{2} \)
17 \( 1 + (-4.5 - 2.59i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.32 - 2.29i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.96 - 2.29i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (1.32 - 2.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.46iT - 41T^{2} \)
43 \( 1 + 9.16iT - 43T^{2} \)
47 \( 1 + (3.96 + 6.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.5 - 2.59i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.5 + 0.866i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.96 - 2.29i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 9.16iT - 71T^{2} \)
73 \( 1 + (4.5 + 2.59i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.96 - 2.29i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.46iT - 97T^{2} \)
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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

−13.70587411356480798883957737158, −12.42538323400478854058334335566, −11.65584053078232224461279358798, −10.49190838752093470044301436626, −8.679649641129702591916026561725, −7.85028033763077792954919297404, −7.28351197987341004820984055941, −5.62059889436959607760730626737, −3.55862285421768153339296382046, −1.88278739554525576338904344952, 2.93861972899947333027840987758, 4.45816054443961004313350621212, 5.20769971486804447228733291836, 7.82789971638098795610452937857, 8.200184996756004186037125806336, 9.666446293111389124783342149200, 10.40759956218351540560249830580, 11.55422542612791787431515353230, 12.75604115994206422946278788981, 14.11521617439140368730256494817

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