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Properties

Label 2-112-112.53-c1-0-0
Degree $2$
Conductor $112$
Sign $0.619 - 0.785i$
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  + (−0.504 − 1.32i)2-s + (0.589 + 2.20i)3-s + (−1.49 + 1.33i)4-s + (−0.622 + 2.32i)5-s + (2.60 − 1.88i)6-s + (−2.41 + 1.09i)7-s + (2.51 + 1.29i)8-s + (−1.89 + 1.09i)9-s + (3.38 − 0.348i)10-s + (0.0284 − 0.00762i)11-s + (−3.81 − 2.49i)12-s + (4.38 − 4.38i)13-s + (2.65 + 2.63i)14-s − 5.47·15-s + (0.450 − 3.97i)16-s + (1.36 − 2.35i)17-s + ⋯
L(s)  = 1  + (−0.356 − 0.934i)2-s + (0.340 + 1.27i)3-s + (−0.745 + 0.666i)4-s + (−0.278 + 1.03i)5-s + (1.06 − 0.770i)6-s + (−0.911 + 0.412i)7-s + (0.888 + 0.459i)8-s + (−0.631 + 0.364i)9-s + (1.06 − 0.110i)10-s + (0.00857 − 0.00229i)11-s + (−1.09 − 0.720i)12-s + (1.21 − 1.21i)13-s + (0.709 + 0.704i)14-s − 1.41·15-s + (0.112 − 0.993i)16-s + (0.330 − 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.619 - 0.785i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.619 - 0.785i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.619 - 0.785i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.750173 + 0.363764i\)
\(L(\frac12)\) \(\approx\) \(0.750173 + 0.363764i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.504 + 1.32i)T \)
7 \( 1 + (2.41 - 1.09i)T \)
good3 \( 1 + (-0.589 - 2.20i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (0.622 - 2.32i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.0284 + 0.00762i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.38 + 4.38i)T - 13iT^{2} \)
17 \( 1 + (-1.36 + 2.35i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.73 - 1.53i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.33 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (4.93 - 4.93i)T - 29iT^{2} \)
31 \( 1 + (1.29 - 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.09 + 7.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 0.207iT - 41T^{2} \)
43 \( 1 + (0.278 + 0.278i)T + 43iT^{2} \)
47 \( 1 + (1.91 + 3.31i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.22 + 0.328i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.208 - 0.0558i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-5.93 - 1.59i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (3.20 + 11.9i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 7.37iT - 71T^{2} \)
73 \( 1 + (3.67 + 2.12i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.51 + 7.81i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.55 - 7.55i)T - 83iT^{2} \)
89 \( 1 + (-3.03 + 1.75i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 11.1T + 97T^{2} \)
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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

−13.76150589691038306223736870872, −12.59160320409587713515270220112, −11.31736137321288045841163390561, −10.50366690549670797826348725005, −9.772977886859533586585630001156, −8.877867525204984302157939808036, −7.47133586629831805951970842546, −5.51637846871348413645559429260, −3.54685214552715820463227541326, −3.20264783132959012111217307944, 1.18947487869370211986090565667, 4.13873251806264673359160340367, 5.96013198832079164294860383274, 6.91061057456994979933300522521, 7.949011449796339574401275325820, 8.818267395360592513371157944625, 9.850191785676275552912452579649, 11.66733910468485187830618758813, 12.94627390542883279849323823876, 13.40396265649831982980218836637

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