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Properties

Label 1-704-704.579-r0-0-0
Degree $1$
Conductor $704$
Sign $0.997 + 0.0678i$
Analytic cond. $3.26936$
Root an. cond. $3.26936$
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  + (−0.522 − 0.852i)3-s + (0.996 − 0.0784i)5-s + (−0.156 + 0.987i)7-s + (−0.453 + 0.891i)9-s + (−0.996 − 0.0784i)13-s + (−0.587 − 0.809i)15-s + (−0.951 − 0.309i)17-s + (0.852 − 0.522i)19-s + (0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (0.987 − 0.156i)25-s + (0.996 − 0.0784i)27-s + (0.972 − 0.233i)29-s + (0.309 + 0.951i)31-s + (−0.0784 + 0.996i)35-s + ⋯
L(s)  = 1  + (−0.522 − 0.852i)3-s + (0.996 − 0.0784i)5-s + (−0.156 + 0.987i)7-s + (−0.453 + 0.891i)9-s + (−0.996 − 0.0784i)13-s + (−0.587 − 0.809i)15-s + (−0.951 − 0.309i)17-s + (0.852 − 0.522i)19-s + (0.923 − 0.382i)21-s + (0.707 + 0.707i)23-s + (0.987 − 0.156i)25-s + (0.996 − 0.0784i)27-s + (0.972 − 0.233i)29-s + (0.309 + 0.951i)31-s + (−0.0784 + 0.996i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(704\)    =    \(2^{6} \cdot 11\)
Sign: $0.997 + 0.0678i$
Analytic conductor: \(3.26936\)
Root analytic conductor: \(3.26936\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{704} (579, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 704,\ (0:\ ),\ 0.997 + 0.0678i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.285096516 + 0.04365296037i\)
\(L(\frac12)\) \(\approx\) \(1.285096516 + 0.04365296037i\)
\(L(1)\) \(\approx\) \(1.019609825 - 0.08966250459i\)
\(L(1)\) \(\approx\) \(1.019609825 - 0.08966250459i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (-0.522 - 0.852i)T \)
5 \( 1 + (0.996 - 0.0784i)T \)
7 \( 1 + (-0.156 + 0.987i)T \)
13 \( 1 + (-0.996 - 0.0784i)T \)
17 \( 1 + (-0.951 - 0.309i)T \)
19 \( 1 + (0.852 - 0.522i)T \)
23 \( 1 + (0.707 + 0.707i)T \)
29 \( 1 + (0.972 - 0.233i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.852 + 0.522i)T \)
41 \( 1 + (-0.156 - 0.987i)T \)
43 \( 1 + (-0.923 + 0.382i)T \)
47 \( 1 + (-0.587 + 0.809i)T \)
53 \( 1 + (0.649 + 0.760i)T \)
59 \( 1 + (0.852 + 0.522i)T \)
61 \( 1 + (0.649 - 0.760i)T \)
67 \( 1 + (0.923 + 0.382i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 + (0.987 + 0.156i)T \)
79 \( 1 + (0.951 - 0.309i)T \)
83 \( 1 + (0.760 + 0.649i)T \)
89 \( 1 + (-0.707 + 0.707i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−22.47727473294807930004474118120, −21.883068952808741274959933147935, −21.08389962724443351619208486582, −20.33210874710277109356649649132, −19.618039070084387279260873821761, −18.22239891747983678297497625054, −17.60028499073705859430313514334, −16.77173130226961695875497592903, −16.42801700162979761010570905681, −15.12617019208987685008849818194, −14.46926893746718169299958262830, −13.5717853212209036651884603555, −12.72582573305161645821260856405, −11.59449923279799767442394624393, −10.72933357023665942202184394650, −9.96145045665997957165768710392, −9.551217682150974647875238284942, −8.35390081534081163709066453200, −6.954357040123109502558502057562, −6.3549357573427085375157307429, −5.19775587982277663548490207955, −4.55285551074377996992373639707, −3.44610754499905851915531625084, −2.29892342129494395744226578026, −0.78170904531938638011442433986, 1.09474400301848647459077680956, 2.29422346350342790032658975964, 2.82640112017739872490469372620, 4.96860541418596213688594834298, 5.31406832783370583900143752956, 6.44760018014581110963444009634, 6.99711581118401875171691611651, 8.23804931512148231818481744592, 9.18142039940542157855820148134, 9.944582187734310908348314810955, 11.12474309600432785142987591774, 11.917552299587387917795062090095, 12.68851642484935809679768972845, 13.43322438367628128272564310079, 14.16423838715097238916651180915, 15.26310787526879288860008545437, 16.18814150818881937831559146658, 17.236078709853600493208770056440, 17.76108267806082917577634939880, 18.36242320701032799666478666388, 19.32135036275080999419965343987, 20.00734406174847495920914396036, 21.267719800086652452250285433137, 22.011626470589236855764513936078, 22.41209511965530934608688933249

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