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Properties

Label 1-896-896.195-r0-0-0
Degree $1$
Conductor $896$
Sign $0.671 - 0.740i$
Analytic cond. $4.16100$
Root an. cond. $4.16100$
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  + (0.195 − 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 − 0.382i)9-s + (0.980 − 0.195i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.555 + 0.831i)27-s + (0.980 + 0.195i)29-s − i·31-s − i·33-s + (0.555 − 0.831i)37-s + ⋯
L(s)  = 1  + (0.195 − 0.980i)3-s + (0.831 + 0.555i)5-s + (−0.923 − 0.382i)9-s + (0.980 − 0.195i)11-s + (−0.831 + 0.555i)13-s + (0.707 − 0.707i)15-s + (0.707 + 0.707i)17-s + (−0.555 − 0.831i)19-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (−0.555 + 0.831i)27-s + (0.980 + 0.195i)29-s − i·31-s − i·33-s + (0.555 − 0.831i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 896 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(896\)    =    \(2^{7} \cdot 7\)
Sign: $0.671 - 0.740i$
Analytic conductor: \(4.16100\)
Root analytic conductor: \(4.16100\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{896} (195, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 896,\ (0:\ ),\ 0.671 - 0.740i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.714116047 - 0.7598153872i\)
\(L(\frac12)\) \(\approx\) \(1.714116047 - 0.7598153872i\)
\(L(1)\) \(\approx\) \(1.291724684 - 0.3329211407i\)
\(L(1)\) \(\approx\) \(1.291724684 - 0.3329211407i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.195 - 0.980i)T \)
5 \( 1 + (0.831 + 0.555i)T \)
11 \( 1 + (0.980 - 0.195i)T \)
13 \( 1 + (-0.831 + 0.555i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (-0.555 - 0.831i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.980 + 0.195i)T \)
31 \( 1 - iT \)
37 \( 1 + (0.555 - 0.831i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.195 - 0.980i)T \)
47 \( 1 + (0.707 + 0.707i)T \)
53 \( 1 + (0.980 - 0.195i)T \)
59 \( 1 + (0.831 + 0.555i)T \)
61 \( 1 + (0.195 - 0.980i)T \)
67 \( 1 + (0.195 - 0.980i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (-0.923 - 0.382i)T \)
79 \( 1 + (-0.707 + 0.707i)T \)
83 \( 1 + (0.555 + 0.831i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 - iT \)
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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

−21.82987268096304316253594038866, −21.37411387728008135553379929795, −20.51794314333453971769457673971, −19.92493728389764237324987416085, −19.09283744060147857255187186065, −17.78671462033234443815618975572, −17.15423935186634708242767761124, −16.59748438459703843779310559836, −15.74040760192767109653478525405, −14.70689876034816254023585075066, −14.258239380795202454219659905077, −13.346455026721284866955902538536, −12.25884748539508486139576991888, −11.60832544073202985171035269902, −10.15550770626497397686444592156, −10.02304537295067451673187704531, −9.04211647405630601610648809649, −8.36714228475753996139769737293, −7.15388976393382103468887848246, −5.953668895424494004608401602334, −5.22146603880238277958415950672, −4.46170411880813147277488944765, −3.38969715033032426600455246961, −2.40607040002186978665253515831, −1.15402392359838062949228205920, 0.970708930736583869461055461256, 2.09273954064499754714696041929, 2.72077149328302504667586060168, 3.95595669796670105681487857224, 5.26827615595234984259878410121, 6.38991877175492336604959442588, 6.66797343394255746369008281314, 7.68525739840328640389987683880, 8.761325723662496580861462632320, 9.43603824699220512298436339358, 10.47586123539862292993211750284, 11.4126001381916365759163980949, 12.25581750866521676338749631951, 13.01387852251914242567309819975, 13.89688045757292334843869975087, 14.51189926552850157505583989972, 15.04003692997460849487253502950, 16.75030882308155882446107277392, 17.09451949600361832302186391133, 17.90978580179219055192023180675, 18.78357950355701443652262480462, 19.32240399673468895001120898886, 20.0508999445364896794692887429, 21.185960162069941671324840330267, 21.872900924963357748052261495886

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