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Properties

Label 2-112-16.5-c1-0-7
Degree $2$
Conductor $112$
Sign $0.999 + 0.00225i$
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  + (−0.411 + 1.35i)2-s + (2.21 − 2.21i)3-s + (−1.66 − 1.11i)4-s + (−0.393 − 0.393i)5-s + (2.08 + 3.90i)6-s + i·7-s + (2.18 − 1.79i)8-s − 6.81i·9-s + (0.693 − 0.370i)10-s + (2.22 + 2.22i)11-s + (−6.14 + 1.21i)12-s + (−3.16 + 3.16i)13-s + (−1.35 − 0.411i)14-s − 1.74·15-s + (1.52 + 3.69i)16-s + 0.980·17-s + ⋯
L(s)  = 1  + (−0.290 + 0.956i)2-s + (1.27 − 1.27i)3-s + (−0.830 − 0.556i)4-s + (−0.175 − 0.175i)5-s + (0.851 + 1.59i)6-s + 0.377i·7-s + (0.774 − 0.633i)8-s − 2.27i·9-s + (0.219 − 0.117i)10-s + (0.671 + 0.671i)11-s + (−1.77 + 0.350i)12-s + (−0.877 + 0.877i)13-s + (−0.361 − 0.109i)14-s − 0.449·15-s + (0.380 + 0.924i)16-s + 0.237·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00225i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00225i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.999 + 0.00225i$
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: \((2,\ 112,\ (\ :1/2),\ 0.999 + 0.00225i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16674 - 0.00131368i\)
\(L(\frac12)\) \(\approx\) \(1.16674 - 0.00131368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.411 - 1.35i)T \)
7 \( 1 - iT \)
good3 \( 1 + (-2.21 + 2.21i)T - 3iT^{2} \)
5 \( 1 + (0.393 + 0.393i)T + 5iT^{2} \)
11 \( 1 + (-2.22 - 2.22i)T + 11iT^{2} \)
13 \( 1 + (3.16 - 3.16i)T - 13iT^{2} \)
17 \( 1 - 0.980T + 17T^{2} \)
19 \( 1 + (5.26 - 5.26i)T - 19iT^{2} \)
23 \( 1 - 1.25iT - 23T^{2} \)
29 \( 1 + (-3.17 + 3.17i)T - 29iT^{2} \)
31 \( 1 + 4.43T + 31T^{2} \)
37 \( 1 + (-0.645 - 0.645i)T + 37iT^{2} \)
41 \( 1 + 1.21iT - 41T^{2} \)
43 \( 1 + (-0.966 - 0.966i)T + 43iT^{2} \)
47 \( 1 - 9.97T + 47T^{2} \)
53 \( 1 + (8.07 + 8.07i)T + 53iT^{2} \)
59 \( 1 + (-1.81 - 1.81i)T + 59iT^{2} \)
61 \( 1 + (2.58 - 2.58i)T - 61iT^{2} \)
67 \( 1 + (1.59 - 1.59i)T - 67iT^{2} \)
71 \( 1 + 0.934iT - 71T^{2} \)
73 \( 1 + 0.710iT - 73T^{2} \)
79 \( 1 - 11.8T + 79T^{2} \)
83 \( 1 + (6.77 - 6.77i)T - 83iT^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 3.03T + 97T^{2} \)
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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

−14.02139658213452372796807146333, −12.72384841065940690645092094055, −12.13026318893189814390630338080, −9.850781368093635943126052877072, −8.934802222399631454900362550843, −8.092117105042642215376255409506, −7.16392909089169193460190136664, −6.24435980469620983877102901180, −4.17448290308061986663734263126, −1.93452806952323018438366983080, 2.71321633398272490540308862750, 3.74396138775656509915848022473, 4.86835062306961281028539354010, 7.57905682084507561028245973455, 8.702368114229735204535232535948, 9.374294621225647799457160937062, 10.47712191456453116688690175680, 11.04958508716006361064412368731, 12.65949144108960157549644472280, 13.73351299836641560604201344974

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