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Properties

Label 2-112-16.5-c1-0-4
Degree $2$
Conductor $112$
Sign $0.829 + 0.557i$
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  + (−1.27 − 0.605i)2-s + (1.39 − 1.39i)3-s + (1.26 + 1.54i)4-s + (2.16 + 2.16i)5-s + (−2.62 + 0.935i)6-s + i·7-s + (−0.681 − 2.74i)8-s − 0.871i·9-s + (−1.45 − 4.07i)10-s + (−3.09 − 3.09i)11-s + (3.91 + 0.391i)12-s + (1.75 − 1.75i)13-s + (0.605 − 1.27i)14-s + 6.02·15-s + (−0.791 + 3.92i)16-s − 5.20·17-s + ⋯
L(s)  = 1  + (−0.903 − 0.428i)2-s + (0.803 − 0.803i)3-s + (0.633 + 0.773i)4-s + (0.968 + 0.968i)5-s + (−1.06 + 0.381i)6-s + 0.377i·7-s + (−0.240 − 0.970i)8-s − 0.290i·9-s + (−0.460 − 1.28i)10-s + (−0.933 − 0.933i)11-s + (1.13 + 0.112i)12-s + (0.486 − 0.486i)13-s + (0.161 − 0.341i)14-s + 1.55·15-s + (−0.197 + 0.980i)16-s − 1.26·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.829 + 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.829 + 0.557i$
Analytic conductor: \(0.894324\)
Root analytic conductor: \(0.945687\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (85, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 112,\ (\ :1/2),\ 0.829 + 0.557i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.928372 - 0.283052i\)
\(L(\frac12)\) \(\approx\) \(0.928372 - 0.283052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.27 + 0.605i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-1.39 + 1.39i)T - 3iT^{2} \)
5 \( 1 + (-2.16 - 2.16i)T + 5iT^{2} \)
11 \( 1 + (3.09 + 3.09i)T + 11iT^{2} \)
13 \( 1 + (-1.75 + 1.75i)T - 13iT^{2} \)
17 \( 1 + 5.20T + 17T^{2} \)
19 \( 1 + (0.851 - 0.851i)T - 19iT^{2} \)
23 \( 1 + 6.15iT - 23T^{2} \)
29 \( 1 + (6.24 - 6.24i)T - 29iT^{2} \)
31 \( 1 + 2.78T + 31T^{2} \)
37 \( 1 + (-4.11 - 4.11i)T + 37iT^{2} \)
41 \( 1 + 6.32iT - 41T^{2} \)
43 \( 1 + (-3.05 - 3.05i)T + 43iT^{2} \)
47 \( 1 - 3.60T + 47T^{2} \)
53 \( 1 + (-5.28 - 5.28i)T + 53iT^{2} \)
59 \( 1 + (7.13 + 7.13i)T + 59iT^{2} \)
61 \( 1 + (-1.03 + 1.03i)T - 61iT^{2} \)
67 \( 1 + (0.966 - 0.966i)T - 67iT^{2} \)
71 \( 1 - 10.0iT - 71T^{2} \)
73 \( 1 + 15.1iT - 73T^{2} \)
79 \( 1 + 6.61T + 79T^{2} \)
83 \( 1 + (7.41 - 7.41i)T - 83iT^{2} \)
89 \( 1 + 3.26iT - 89T^{2} \)
97 \( 1 + 7.66T + 97T^{2} \)
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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

−13.34121045635478565039560317398, −12.74920259100619369303801641910, −10.97945754217153553912700322712, −10.54462886067023117615568408774, −9.055237527701256912654368629981, −8.282837338989729770745466932504, −7.12580118620505610700957926165, −6.03160710142085535713867559366, −2.99904967314338074963587456243, −2.15460867979968697916247934910, 2.09055935893573580568223504325, 4.45887692388295544178784924984, 5.81155618748778068886143249812, 7.39199563125755620943118252369, 8.696759617365539933064223524408, 9.396464650514175196968139841677, 10.00739035157310388853402854782, 11.23736040617645082297903704101, 13.01649011729891511947310364413, 13.81999090686044216263730349489

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