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Properties

Label 1-847-847.101-r0-0-0
Degree $1$
Conductor $847$
Sign $0.193 - 0.981i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (−0.988 − 0.151i)2-s + (0.978 − 0.207i)3-s + (0.953 + 0.299i)4-s + (0.969 − 0.244i)5-s + (−0.998 + 0.0570i)6-s + (−0.897 − 0.441i)8-s + (0.913 − 0.406i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (0.820 + 0.572i)16-s + (−0.991 − 0.132i)17-s + (−0.964 + 0.263i)18-s + (−0.432 − 0.901i)19-s + (0.998 + 0.0570i)20-s + ⋯
L(s)  = 1  + (−0.988 − 0.151i)2-s + (0.978 − 0.207i)3-s + (0.953 + 0.299i)4-s + (0.969 − 0.244i)5-s + (−0.998 + 0.0570i)6-s + (−0.897 − 0.441i)8-s + (0.913 − 0.406i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.198 − 0.980i)13-s + (0.897 − 0.441i)15-s + (0.820 + 0.572i)16-s + (−0.991 − 0.132i)17-s + (−0.964 + 0.263i)18-s + (−0.432 − 0.901i)19-s + (0.998 + 0.0570i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.193 - 0.981i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.193 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.190828875 - 0.9793761721i\)
\(L(\frac12)\) \(\approx\) \(1.190828875 - 0.9793761721i\)
\(L(1)\) \(\approx\) \(1.067407335 - 0.3647650495i\)
\(L(1)\) \(\approx\) \(1.067407335 - 0.3647650495i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.988 - 0.151i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (0.969 - 0.244i)T \)
13 \( 1 + (0.198 - 0.980i)T \)
17 \( 1 + (-0.991 - 0.132i)T \)
19 \( 1 + (-0.432 - 0.901i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (0.0285 + 0.999i)T \)
31 \( 1 + (-0.861 - 0.508i)T \)
37 \( 1 + (-0.595 + 0.803i)T \)
41 \( 1 + (0.774 - 0.633i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.964 - 0.263i)T \)
53 \( 1 + (0.820 - 0.572i)T \)
59 \( 1 + (-0.161 + 0.986i)T \)
61 \( 1 + (-0.625 - 0.780i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (0.941 - 0.336i)T \)
73 \( 1 + (0.123 + 0.992i)T \)
79 \( 1 + (0.532 + 0.846i)T \)
83 \( 1 + (0.974 + 0.226i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (-0.696 + 0.717i)T \)
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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

−21.84181629859097714395810169211, −21.26120769163710827666871908064, −20.7076873428444388362492912383, −19.66728643350115596758670704533, −19.184093036649349951175710845281, −18.2570413871700900217996949491, −17.68887103583160677137494332320, −16.65541879907047796367100250179, −15.99303793538875455819661023546, −15.03077690523525046893682784434, −14.350141773341974479459111565870, −13.61140441213567484868945189670, −12.630172560079943569438917620838, −11.33408423722162353333619775315, −10.547111040275757499516471560956, −9.66525489225789555034287231283, −9.20663804682482488347556483940, −8.39305281798626206268819617613, −7.439178462226565367289450714091, −6.58353823864500320366910212504, −5.755363310792756418073046416610, −4.33068579240556610238737283262, −3.147494807575765386609164910921, −2.0365129864829458897700431919, −1.64394854759553557129052709504, 0.814357443288016097728088666390, 2.032438522941969613570390218686, 2.54907883977394656503226506557, 3.65472375622603460947394771023, 5.077555606355147927863254514218, 6.37375493727098837197956943437, 7.00012667834295810170867857278, 8.110166152168899435840269416945, 8.81918693221625210790674473597, 9.33723272246257654099756145225, 10.331873573791707056200085205667, 10.894519619638446702287546679838, 12.3573937720098116511239857682, 12.97628783743796031345010921959, 13.75495794123595043154388942299, 14.84391498220029340462774136714, 15.50084060604929620500846829939, 16.43295612871967923877141867481, 17.39419234255681945613221524754, 18.07965462822082984575869449921, 18.57244033754836414771550540786, 19.759036951341801553381044706246, 20.154469218478841077620720727185, 20.85800149939534184702082225537, 21.6271691255549862974833586278

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