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Properties

Label 2-112-112.37-c1-0-2
Degree $2$
Conductor $112$
Sign $0.983 - 0.179i$
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  + (−0.892 + 1.09i)2-s + (−2.20 − 0.589i)3-s + (−0.407 − 1.95i)4-s + (2.32 − 0.622i)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 + (−1.38 + 3.10i)10-s + (−0.00762 + 0.0284i)11-s + (−0.257 + 4.54i)12-s + (4.38 − 4.38i)13-s + (−3.34 + 1.67i)14-s − 5.47·15-s + (−3.66 + 1.59i)16-s + (1.36 + 2.35i)17-s + ⋯
L(s)  = 1  + (−0.630 + 0.775i)2-s + (−1.27 − 0.340i)3-s + (−0.203 − 0.978i)4-s + (1.03 − 0.278i)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 + (−0.439 + 0.981i)10-s + (−0.00229 + 0.00857i)11-s + (−0.0742 + 1.31i)12-s + (1.21 − 1.21i)13-s + (−0.894 + 0.446i)14-s − 1.41·15-s + (−0.916 + 0.399i)16-s + (0.330 + 0.571i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.983 - 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.665103 + 0.0600567i\)
\(L(\frac12)\) \(\approx\) \(0.665103 + 0.0600567i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.892 - 1.09i)T \)
7 \( 1 + (-2.41 - 1.09i)T \)
good3 \( 1 + (2.20 + 0.589i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-2.32 + 0.622i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.00762 - 0.0284i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-4.38 + 4.38i)T - 13iT^{2} \)
17 \( 1 + (-1.36 - 2.35i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.53 + 5.73i)T + (-16.4 + 9.5i)T^{2} \)
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 \( 1 + (7.83 - 2.09i)T + (32.0 - 18.5i)T^{2} \)
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 \( 1 + (0.328 - 1.22i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-0.0558 + 0.208i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.59 + 5.93i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-11.9 - 3.20i)T + (58.0 + 33.5i)T^{2} \)
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

−13.61183813156286229110270999027, −12.74559253430636468460784836041, −11.15627332206774219453781625605, −10.70772461165903774075449497565, −9.251926521639312599976602098880, −8.226343730749998308319529322064, −6.76459024910986098419699839616, −5.64149496710124893803268983495, −5.24265945426836201566307489644, −1.37366231507201300036380420308, 1.68635429890976212326253207329, 4.11063758742151773389233286486, 5.55075868073851709368799300313, 6.84015790591474842683625426799, 8.451214988332150021933880693767, 9.726544661977663316972605229983, 10.63984569047658512494210481049, 11.25639197065294482073486200025, 12.10638413689504621441906276286, 13.50192293505017683269243010125

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