Properties

Label 1-847-847.601-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: $\chi_{847} (601, \cdot )$
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

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

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