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Properties

Label 2-112-112.83-c1-0-2
Degree $2$
Conductor $112$
Sign $0.964 - 0.264i$
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 + 0.5i)2-s + (1.50 − 1.32i)4-s + 2.64·7-s + (−1.32 + 2.50i)8-s − 3i·9-s + (4.64 + 4.64i)11-s + (−3.50 + 1.32i)14-s + (0.500 − 3.96i)16-s + (1.5 + 3.96i)18-s + (−8.46 − 3.82i)22-s − 8·23-s − 5i·25-s + (3.96 − 3.50i)28-s + (−4.29 − 4.29i)29-s + (1.32 + 5.50i)32-s + ⋯
L(s)  = 1  + (−0.935 + 0.353i)2-s + (0.750 − 0.661i)4-s + 0.999·7-s + (−0.467 + 0.883i)8-s − i·9-s + (1.40 + 1.40i)11-s + (−0.935 + 0.353i)14-s + (0.125 − 0.992i)16-s + (0.353 + 0.935i)18-s + (−1.80 − 0.815i)22-s − 1.66·23-s − i·25-s + (0.749 − 0.661i)28-s + (−0.796 − 0.796i)29-s + (0.233 + 0.972i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.964 - 0.264i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.964 - 0.264i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.964 - 0.264i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.779537 + 0.104838i\)
\(L(\frac12)\) \(\approx\) \(0.779537 + 0.104838i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.32 - 0.5i)T \)
7 \( 1 - 2.64T \)
good3 \( 1 + 3iT^{2} \)
5 \( 1 + 5iT^{2} \)
11 \( 1 + (-4.64 - 4.64i)T + 11iT^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19iT^{2} \)
23 \( 1 + 8T + 23T^{2} \)
29 \( 1 + (4.29 + 4.29i)T + 29iT^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (8.29 - 8.29i)T - 37iT^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + (-3.35 - 3.35i)T + 43iT^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + (-0.291 + 0.291i)T - 53iT^{2} \)
59 \( 1 - 59iT^{2} \)
61 \( 1 - 61iT^{2} \)
67 \( 1 + (5.93 - 5.93i)T - 67iT^{2} \)
71 \( 1 + 5.29T + 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 + 8iT - 79T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−14.15327935312368531648856410960, −12.08326454902218118248382877410, −11.73769192336234030652554216834, −10.20931246479493097671327371609, −9.388457229823937646020779570282, −8.310202059027090035321052854047, −7.13597810339420375884779245306, −6.09843626077197207747919334374, −4.31814910027189803742301917561, −1.73815702022737060387468509209, 1.75059686069270978242917182210, 3.77686020842783578524989495642, 5.73476959588001284074065986455, 7.30320941447478430103971355842, 8.337932224675408309894902133446, 9.144358280335224860950406053715, 10.64081692189038561800590858750, 11.26302580361987185724281338319, 12.14689701557299283862467982082, 13.66550408696699066261985184020

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