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Properties

Label 1-847-847.158-r0-0-0
Degree $1$
Conductor $847$
Sign $0.367 - 0.930i$
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  + (−0.217 − 0.976i)2-s + (−0.104 + 0.994i)3-s + (−0.905 + 0.424i)4-s + (−0.851 − 0.524i)5-s + (0.993 − 0.113i)6-s + (0.610 + 0.791i)8-s + (−0.978 − 0.207i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (−0.921 − 0.389i)13-s + (0.610 − 0.791i)15-s + (0.640 − 0.768i)16-s + (−0.710 + 0.703i)17-s + (0.00951 + 0.999i)18-s + (0.988 + 0.151i)19-s + (0.993 + 0.113i)20-s + ⋯
L(s)  = 1  + (−0.217 − 0.976i)2-s + (−0.104 + 0.994i)3-s + (−0.905 + 0.424i)4-s + (−0.851 − 0.524i)5-s + (0.993 − 0.113i)6-s + (0.610 + 0.791i)8-s + (−0.978 − 0.207i)9-s + (−0.327 + 0.945i)10-s + (−0.327 − 0.945i)12-s + (−0.921 − 0.389i)13-s + (0.610 − 0.791i)15-s + (0.640 − 0.768i)16-s + (−0.710 + 0.703i)17-s + (0.00951 + 0.999i)18-s + (0.988 + 0.151i)19-s + (0.993 + 0.113i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.367 - 0.930i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.367 - 0.930i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.367 - 0.930i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5178050940 - 0.3521712008i\)
\(L(\frac12)\) \(\approx\) \(0.5178050940 - 0.3521712008i\)
\(L(1)\) \(\approx\) \(0.6214945650 - 0.1545050850i\)
\(L(1)\) \(\approx\) \(0.6214945650 - 0.1545050850i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.217 - 0.976i)T \)
3 \( 1 + (-0.104 + 0.994i)T \)
5 \( 1 + (-0.851 - 0.524i)T \)
13 \( 1 + (-0.921 - 0.389i)T \)
17 \( 1 + (-0.710 + 0.703i)T \)
19 \( 1 + (0.988 + 0.151i)T \)
23 \( 1 + (0.928 + 0.371i)T \)
29 \( 1 + (-0.998 + 0.0570i)T \)
31 \( 1 + (-0.999 - 0.0190i)T \)
37 \( 1 + (-0.683 + 0.730i)T \)
41 \( 1 + (0.198 - 0.980i)T \)
43 \( 1 + (-0.959 - 0.281i)T \)
47 \( 1 + (0.00951 - 0.999i)T \)
53 \( 1 + (0.640 + 0.768i)T \)
59 \( 1 + (0.749 - 0.662i)T \)
61 \( 1 + (0.953 - 0.299i)T \)
67 \( 1 + (0.580 - 0.814i)T \)
71 \( 1 + (0.774 - 0.633i)T \)
73 \( 1 + (0.272 - 0.962i)T \)
79 \( 1 + (0.997 - 0.0760i)T \)
83 \( 1 + (0.897 + 0.441i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
97 \( 1 + (-0.0285 - 0.999i)T \)
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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

−22.63534385763964943887064287976, −21.93548412914827825673621332458, −20.28375247784947234650774469482, −19.57099690722814915998309892442, −18.888594017845398949550864948661, −18.24365993758754635449603479964, −17.57919770252069558919789178374, −16.61080556836651960520160069395, −15.96815551468087053293742144421, −14.82329051290761109550627395626, −14.46567959920070748942123570550, −13.443487780364655609656894298305, −12.690448390474366965605140733091, −11.65654912949752327076097769594, −10.98969497240698803940838742595, −9.65617622200466187074047556414, −8.79183597866246396183799277328, −7.86054399816377614066078248564, −7.0330535011530219593197646534, −6.91200469607876598867929139606, −5.558114594013004520515671021722, −4.74399562971776449839529150736, −3.48301343317268771051573885399, −2.291863885690207108215907220621, −0.77486454188393315263518611060, 0.46799872422021030517000611953, 2.01471375306268215317475117345, 3.36108939311917294547874936837, 3.783939341930953785075469785083, 4.94766209519725588845376103491, 5.32957533118743833022221147200, 7.20673263402926075601369705123, 8.18243305673903143721869530798, 8.979909832499306398554927065113, 9.62913105189978165134894142893, 10.5890322019316338595036966417, 11.26482887363852521451799236489, 12.00330973261340997525272989779, 12.757938129128385206961760962970, 13.74214934803137296902287554266, 14.91033331895202259943769381479, 15.412876795229225841226087869082, 16.61976428706764791458757088221, 17.020829547935150305297405549793, 17.97734145313559638940706076748, 19.08108511273672413591803441584, 19.81368617046647914952819804703, 20.33851794700756357747804017103, 20.975048476186143748398715923016, 22.0426262650820712378618655441

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