Properties

Label 1-847-847.272-r0-0-0
Degree $1$
Conductor $847$
Sign $0.944 + 0.328i$
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  + (−0.993 + 0.113i)2-s + (0.809 − 0.587i)3-s + (0.974 − 0.226i)4-s + (0.564 + 0.825i)5-s + (−0.736 + 0.676i)6-s + (−0.941 + 0.336i)8-s + (0.309 − 0.951i)9-s + (−0.654 − 0.755i)10-s + (0.654 − 0.755i)12-s + (0.516 + 0.856i)13-s + (0.941 + 0.336i)15-s + (0.897 − 0.441i)16-s + (0.774 + 0.633i)17-s + (−0.198 + 0.980i)18-s + (−0.998 + 0.0570i)19-s + (0.736 + 0.676i)20-s + ⋯
L(s)  = 1  + (−0.993 + 0.113i)2-s + (0.809 − 0.587i)3-s + (0.974 − 0.226i)4-s + (0.564 + 0.825i)5-s + (−0.736 + 0.676i)6-s + (−0.941 + 0.336i)8-s + (0.309 − 0.951i)9-s + (−0.654 − 0.755i)10-s + (0.654 − 0.755i)12-s + (0.516 + 0.856i)13-s + (0.941 + 0.336i)15-s + (0.897 − 0.441i)16-s + (0.774 + 0.633i)17-s + (−0.198 + 0.980i)18-s + (−0.998 + 0.0570i)19-s + (0.736 + 0.676i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.328i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.944 + 0.328i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.944 + 0.328i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.442076241 + 0.2437805658i\)
\(L(\frac12)\) \(\approx\) \(1.442076241 + 0.2437805658i\)
\(L(1)\) \(\approx\) \(1.062237089 + 0.04933445023i\)
\(L(1)\) \(\approx\) \(1.062237089 + 0.04933445023i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.993 + 0.113i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.564 + 0.825i)T \)
13 \( 1 + (0.516 + 0.856i)T \)
17 \( 1 + (0.774 + 0.633i)T \)
19 \( 1 + (-0.998 + 0.0570i)T \)
23 \( 1 + (-0.142 - 0.989i)T \)
29 \( 1 + (0.362 + 0.931i)T \)
31 \( 1 + (0.921 - 0.389i)T \)
37 \( 1 + (0.0855 + 0.996i)T \)
41 \( 1 + (-0.870 - 0.491i)T \)
43 \( 1 + (0.959 + 0.281i)T \)
47 \( 1 + (-0.198 - 0.980i)T \)
53 \( 1 + (0.897 + 0.441i)T \)
59 \( 1 + (0.870 - 0.491i)T \)
61 \( 1 + (0.993 + 0.113i)T \)
67 \( 1 + (0.415 + 0.909i)T \)
71 \( 1 + (-0.254 + 0.967i)T \)
73 \( 1 + (-0.466 + 0.884i)T \)
79 \( 1 + (0.0285 - 0.999i)T \)
83 \( 1 + (-0.985 + 0.170i)T \)
89 \( 1 + (-0.841 + 0.540i)T \)
97 \( 1 + (0.564 - 0.825i)T \)
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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

−21.47525579296907175316554766420, −21.09081167984998976655983944004, −20.507125395697032525093901022203, −19.6617266693696142785313180414, −19.102962784605237556994839232290, −17.97541722765010740192144690440, −17.31652115835832040870736371931, −16.43954489760416094052265613237, −15.80790530436081150583755604585, −15.098184515897188853163341617934, −13.994945876740899967323986311565, −13.159686341429314603264012124236, −12.286614116435798915835293197128, −11.178133433687039448679460367, −10.17312440164992852043686010333, −9.749812291535971543076678209495, −8.83113010282905206805511205946, −8.25365594187968445543399225286, −7.487932171372573322125196025882, −6.13000423932528972927056346541, −5.2344288318377418008286159215, −4.00311020634478713758017446102, −2.94117755641705764485443615734, −2.043312144915211782602953493688, −0.93621644037154574291558749621, 1.224698060976063822220558177159, 2.10544047990358912436576190229, 2.85895632417346259702296054071, 3.951337032357223294610656864602, 5.80798504429858383365418751817, 6.63464016376224908740719258286, 7.05468282093717709179239099457, 8.323593263577888350498718344257, 8.674182615326475968240753856573, 9.85661440484845089516314500041, 10.35243579004465247242903836850, 11.41089520188589750212773973162, 12.32904001395186747059938960774, 13.340486619860202246631436557068, 14.38082770522018764009817177992, 14.73537991593970438159258066650, 15.704510102280757379408416549533, 16.79777112276645176880544644438, 17.49943567752982160174674576797, 18.44816559795346052614109160206, 18.84382335064780146141017867816, 19.40460398475803552046808330199, 20.49781636571977382716472346832, 21.09535314825777807229972153504, 21.85794171639856976401660297905

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