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Properties

Label 2-112-112.109-c1-0-13
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: $\chi_{112} (109, \cdot )$
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

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

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