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Properties

Label 1-56-56.5-r1-0-0
Degree $1$
Conductor $56$
Sign $0.386 - 0.922i$
Analytic cond. $6.01803$
Root an. cond. $6.01803$
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  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.5 − 0.866i)5-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + 13-s + 15-s + (0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + 27-s − 29-s + (0.5 − 0.866i)31-s + (0.5 + 0.866i)33-s + (0.5 + 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(56\)    =    \(2^{3} \cdot 7\)
Sign: $0.386 - 0.922i$
Analytic conductor: \(6.01803\)
Root analytic conductor: \(6.01803\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{56} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 56,\ (1:\ ),\ 0.386 - 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8207880187 - 0.5459733256i\)
\(L(\frac12)\) \(\approx\) \(0.8207880187 - 0.5459733256i\)
\(L(1)\) \(\approx\) \(0.8328904045 - 0.1061381970i\)
\(L(1)\) \(\approx\) \(0.8328904045 - 0.1061381970i\)

Euler product

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

−33.303190839132080343342023080206, −31.54426221373143704525488251646, −30.396777748075247186238429915092, −29.97096474977585039755147784180, −28.42051650161422327708627593285, −27.56459954820951227424890161982, −25.978032504612118492662790419023, −25.092500195710263402839434735537, −23.47629534286427331197175326959, −23.05956352751087535748522071738, −21.75718188401715853439926326543, −19.98578246234125872338731830738, −18.90313038073641507621087051328, −18.0279745963835417083146788612, −16.79708786168985219741984886888, −15.23698815476838499339514135274, −14.003169338092966937197428270498, −12.55658441379179326952439002714, −11.50197194818431498963583880353, −10.31464188653018778364807097620, −8.20704584444199725366806153043, −7.028480989586577041650385685318, −5.898906527966484116955254070158, −3.76052958575870632124191886021, −1.73250857361019227909332214378, 0.61143003446968288555295849038, 3.58091852871631669829789861368, 4.8293321138044502499171325478, 6.22115016599222491301287011002, 8.354611778351419843284759161523, 9.40368565478028008471827981517, 11.03368982274204631596498235210, 11.93023590152752959682407220236, 13.52409807795634397353438546019, 15.17237094910416958100298558312, 16.267694875051908751783799453753, 16.94989854042810753062585736632, 18.59244678460921910002699420201, 20.14890543190247601488888469434, 21.000356189615498531901517442716, 22.22008816712626582141357215530, 23.38064085557850779850195030591, 24.400984912708843331777041346336, 25.92777454988139201417122187758, 27.20106867896025280707143296272, 27.93744762664238255381557106643, 28.87334563532176514710159460243, 30.27636903208698263528280715912, 31.89571297118287268128528324895, 32.36506566043854788882292500724

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