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Properties

Label 1-112-112.75-r0-0-0
Degree $1$
Conductor $112$
Sign $0.496 + 0.868i$
Analytic cond. $0.520125$
Root an. cond. $0.520125$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.866 + 0.5i)5-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s − i·13-s − 15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s − i·29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.866 − 0.5i)37-s + ⋯
L(s)  = 1  + (0.866 + 0.5i)3-s + (−0.866 + 0.5i)5-s + (0.5 + 0.866i)9-s + (0.866 + 0.5i)11-s − i·13-s − 15-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (−0.5 − 0.866i)23-s + (0.5 − 0.866i)25-s − i·27-s − i·29-s + (−0.5 + 0.866i)31-s + (0.5 + 0.866i)33-s + (0.866 − 0.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.496 + 0.868i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.496 + 0.868i$
Analytic conductor: \(0.520125\)
Root analytic conductor: \(0.520125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 112,\ (0:\ ),\ 0.496 + 0.868i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.045172896 + 0.6063254759i\)
\(L(\frac12)\) \(\approx\) \(1.045172896 + 0.6063254759i\)
\(L(1)\) \(\approx\) \(1.142765621 + 0.3694228632i\)
\(L(1)\) \(\approx\) \(1.142765621 + 0.3694228632i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 - iT \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.866 + 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.866 - 0.5i)T \)
59 \( 1 + (-0.866 - 0.5i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 - iT \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 - T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−29.59317839064434081270714031774, −27.91620222042582887325717661434, −27.33900418410715530842319253412, −26.057248610418357271989875952559, −25.14059802402527056087186532755, −24.1339443785468601467974586758, −23.44398557397912022473453841248, −21.97544548544072555333963040389, −20.72462558797610550289465326940, −19.68923923724423683558811259938, −19.277402000591513622663638012306, −17.87473664114702248904928951566, −16.61521027312850370302227445488, −15.33027425453244559338560723108, −14.55528433307729987408756103025, −13.14504636345987326441909167326, −12.390071218909455045643951876909, −11.10138944506069174676862317826, −9.42097951715472858684139307297, −8.360850916414274752366541969366, −7.58125641609044677094344205254, −6.082724903957280900383158717246, −4.2015385306709965453943741682, −3.15388290689385467547881479181, −1.26987235759814712816482740371, 2.20028912059130880753082369638, 3.69268631992484908630145979973, 4.51512859866294903692999089966, 6.65502067331196947474403981907, 7.764368789031816981215912719876, 8.928705636993300521711307624836, 10.03276237014593977608140745592, 11.31234078098549486904457411463, 12.44501229612405798734701320463, 14.15775287580042514083014702996, 14.64010036994408425639945126288, 15.82695963618144056485850808037, 16.7321347644779983834942278056, 18.49036406528103816330380402837, 19.32244381538837035664925743481, 20.17903031870114468598521976277, 21.26769289795776068772244074390, 22.36870794057568245175590531219, 23.34157104528604441173595638964, 24.66185934125923468191633302111, 25.62098554832790495128912161942, 26.642183051247428793368479991588, 27.320660413939962524099503933180, 28.27529797487378860315782985572, 29.91952429238476870795380360582

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