Properties

Label 1-847-847.369-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: $\chi_{847} (369, \cdot )$
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

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

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