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Properties

Label 1-112-112.53-r0-0-0
Degree $1$
Conductor $112$
Sign $0.557 - 0.830i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(112\)    =    \(2^{4} \cdot 7\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(0.520125\)
Root analytic conductor: \(0.520125\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{112} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 112,\ (0:\ ),\ 0.557 - 0.830i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5613111577 - 0.2992788649i\)
\(L(\frac12)\) \(\approx\) \(0.5613111577 - 0.2992788649i\)
\(L(1)\) \(\approx\) \(0.7095540519 - 0.1068698610i\)
\(L(1)\) \(\approx\) \(0.7095540519 - 0.1068698610i\)

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

−29.55338305415946101522421822418, −28.526361692282611003600051002612, −27.644614265384369630423320647231, −26.70211222019814729098657363546, −25.43710131229158947896942226414, −24.10806290053508461402772948087, −23.53861358293744406788995634227, −22.43024384198286127196622979748, −21.74610244241076314785809508069, −19.92193375926294571279567896520, −19.173144005096466791374122735147, −18.1131363573024500689922851312, −17.09040082752336298912758358072, −16.02491276649183154734434452782, −14.88036838358094607733425211702, −13.57350815873539042991517588784, −12.15451439880451917911102186988, −11.55300316921499673431913340582, −10.46638543371874974513546403674, −8.83241074067976284184113160010, −7.21477088972652191889870981283, −6.6917311382954898883846470, −5.01646831488594786411824541743, −3.69244623396000764866059264595, −1.63827248548557346591775499427, 0.752599787043619165167188340425, 3.436077675758855146119301048217, 4.598077998491130869935779234375, 5.759243625358865538625837186616, 7.18107992042351805095300174756, 8.60991348364358459668724758099, 9.83920564591008654107735479886, 11.22225202303983468152549704300, 11.880191887997559424986704158, 13.046381663937522038783829760794, 14.72513459552023444808801334529, 15.81765918818891906605109419007, 16.535849737575948734950478136089, 17.59532552811897080479862764646, 18.81347554617034825615488480892, 20.11593971830440120740586837275, 20.92292451511616567342793258640, 22.49775001077533924611351932315, 22.74409261274972338031917345775, 24.16351765817277017663995388274, 24.85248455718055030184604032911, 26.70218248793482303582507004349, 27.26653618724754919913991319880, 28.10424304317487697696065279334, 29.10394338519964561161345806465

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