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Properties

Label 2-112-112.37-c1-0-1
Degree $2$
Conductor $112$
Sign $-0.998 - 0.0599i$
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  + (0.713 + 1.22i)2-s + (−3.04 − 0.814i)3-s + (−0.980 + 1.74i)4-s + (−1.87 + 0.501i)5-s + (−1.17 − 4.29i)6-s + (−1.89 + 1.84i)7-s + (−2.82 + 0.0473i)8-s + (5.98 + 3.45i)9-s + (−1.94 − 1.92i)10-s + (0.299 − 1.11i)11-s + (4.40 − 4.50i)12-s + (0.00680 − 0.00680i)13-s + (−3.60 − 1.00i)14-s + 6.10·15-s + (−2.07 − 3.41i)16-s + (1.52 + 2.63i)17-s + ⋯
L(s)  = 1  + (0.504 + 0.863i)2-s + (−1.75 − 0.470i)3-s + (−0.490 + 0.871i)4-s + (−0.837 + 0.224i)5-s + (−0.480 − 1.75i)6-s + (−0.717 + 0.696i)7-s + (−0.999 + 0.0167i)8-s + (1.99 + 1.15i)9-s + (−0.616 − 0.609i)10-s + (0.0904 − 0.337i)11-s + (1.27 − 1.29i)12-s + (0.00188 − 0.00188i)13-s + (−0.963 − 0.267i)14-s + 1.57·15-s + (−0.519 − 0.854i)16-s + (0.369 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0599i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $-0.998 - 0.0599i$
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: \((2,\ 112,\ (\ :1/2),\ -0.998 - 0.0599i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0110309 + 0.367681i\)
\(L(\frac12)\) \(\approx\) \(0.0110309 + 0.367681i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.713 - 1.22i)T \)
7 \( 1 + (1.89 - 1.84i)T \)
good3 \( 1 + (3.04 + 0.814i)T + (2.59 + 1.5i)T^{2} \)
5 \( 1 + (1.87 - 0.501i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.299 + 1.11i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-0.00680 + 0.00680i)T - 13iT^{2} \)
17 \( 1 + (-1.52 - 2.63i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.24 - 4.64i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (4.27 + 2.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.45 - 1.45i)T - 29iT^{2} \)
31 \( 1 + (-2.60 - 4.51i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (8.51 - 2.28i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 6.02iT - 41T^{2} \)
43 \( 1 + (-7.17 - 7.17i)T + 43iT^{2} \)
47 \( 1 + (-0.796 + 1.38i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.211 + 0.787i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.94 + 7.25i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (-2.46 - 9.21i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-6.35 - 1.70i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 6.77iT - 71T^{2} \)
73 \( 1 + (-3.43 + 1.98i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.81 - 4.87i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.3 - 11.3i)T - 83iT^{2} \)
89 \( 1 + (3.59 + 2.07i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.390T + 97T^{2} \)
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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

−14.13363721128786543281452331497, −12.69307898057556772440171013160, −12.27192876404382754077868107916, −11.47302745029259549022865346187, −10.12264731988518566499975461638, −8.308613905319174835529306564364, −7.13295520840581351245699238624, −6.16869792803696354971133815418, −5.41628564241033779905610555412, −3.83372218572071538449543643367, 0.42849374404219064707418476199, 3.80900358098968765310894486659, 4.72148653529427131140659528810, 5.93592498120428332957092544217, 7.19344092405287603296453825147, 9.495654037236077574680977679474, 10.26837856830352655978488146672, 11.28351180613848875782227052003, 11.89579427009588627719729681956, 12.68585644381975842496687074291

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