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Properties

Label 2-112-112.37-c1-0-8
Degree $2$
Conductor $112$
Sign $-0.101 + 0.994i$
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  + (−1.36 − 0.381i)2-s + (−1.60 − 0.430i)3-s + (1.70 + 1.03i)4-s + (1.94 − 0.522i)5-s + (2.02 + 1.20i)6-s + (−1.89 − 1.85i)7-s + (−1.93 − 2.06i)8-s + (−0.196 − 0.113i)9-s + (−2.85 − 0.0319i)10-s + (1.63 − 6.09i)11-s + (−2.30 − 2.40i)12-s + (1.13 − 1.13i)13-s + (1.86 + 3.24i)14-s − 3.35·15-s + (1.84 + 3.55i)16-s + (−0.960 − 1.66i)17-s + ⋯
L(s)  = 1  + (−0.962 − 0.269i)2-s + (−0.928 − 0.248i)3-s + (0.854 + 0.519i)4-s + (0.871 − 0.233i)5-s + (0.827 + 0.490i)6-s + (−0.714 − 0.699i)7-s + (−0.682 − 0.730i)8-s + (−0.0655 − 0.0378i)9-s + (−0.902 − 0.0101i)10-s + (0.492 − 1.83i)11-s + (−0.664 − 0.694i)12-s + (0.314 − 0.314i)13-s + (0.499 + 0.866i)14-s − 0.867·15-s + (0.460 + 0.887i)16-s + (−0.233 − 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.101 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.101 + 0.994i$
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: \((2,\ 112,\ (\ :1/2),\ -0.101 + 0.994i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.349377 - 0.386824i\)
\(L(\frac12)\) \(\approx\) \(0.349377 - 0.386824i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.36 + 0.381i)T \)
7 \( 1 + (1.89 + 1.85i)T \)
good3 \( 1 + (1.60 + 0.430i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (-1.94 + 0.522i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.63 + 6.09i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.13 + 1.13i)T - 13iT^{2} \)
17 \( 1 + (0.960 + 1.66i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.63 - 6.09i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.924 + 0.533i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.08 + 5.08i)T - 29iT^{2} \)
31 \( 1 + (0.198 + 0.343i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.327 + 0.0877i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 7.26iT - 41T^{2} \)
43 \( 1 + (-1.75 - 1.75i)T + 43iT^{2} \)
47 \( 1 + (1.08 - 1.87i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.111 + 0.415i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (2.32 - 8.67i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.72 + 10.1i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-7.87 - 2.10i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 2.78iT - 71T^{2} \)
73 \( 1 + (-2.05 + 1.18i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-5.10 + 8.84i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-1.17 + 1.17i)T - 83iT^{2} \)
89 \( 1 + (-11.0 - 6.37i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.21T + 97T^{2} \)
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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

−13.20218574720820875570515910412, −12.01951612460559255970138440808, −11.15090969168084391105740374634, −10.23019135126167079146829031812, −9.240577640005565461125479145423, −8.025174549114710229466500294200, −6.34498697103080927555736325087, −5.96451651277535007446635468329, −3.33831491049437256000268987635, −0.901714340244421763534196521492, 2.25500096370910710768764165627, 5.09671595318780222199091983001, 6.26266551010151639255438064134, 6.98548654620016571630387847264, 8.870253984949308659516763823176, 9.705270664756684309969187990450, 10.49834751828140535456224845259, 11.65171549831466274733658751452, 12.53343305712420806434283180969, 14.08309865827864255817784522800

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