Properties

Label 1-847-847.545-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.531 - 0.846i$
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  + (0.998 − 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (0.466 − 0.884i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (−0.809 + 0.587i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (−0.774 + 0.633i)18-s + (−0.0285 − 0.999i)19-s + (0.362 − 0.931i)20-s + ⋯
L(s)  = 1  + (0.998 − 0.0570i)2-s + (−0.309 − 0.951i)3-s + (0.993 − 0.113i)4-s + (0.466 − 0.884i)5-s + (−0.362 − 0.931i)6-s + (0.985 − 0.170i)8-s + (−0.809 + 0.587i)9-s + (0.415 − 0.909i)10-s + (−0.415 − 0.909i)12-s + (−0.870 − 0.491i)13-s + (−0.985 − 0.170i)15-s + (0.974 − 0.226i)16-s + (0.941 + 0.336i)17-s + (−0.774 + 0.633i)18-s + (−0.0285 − 0.999i)19-s + (0.362 − 0.931i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.531 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.531 - 0.846i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.531 - 0.846i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.246063463 - 2.254088339i\)
\(L(\frac12)\) \(\approx\) \(1.246063463 - 2.254088339i\)
\(L(1)\) \(\approx\) \(1.524177143 - 1.009022392i\)
\(L(1)\) \(\approx\) \(1.524177143 - 1.009022392i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.998 - 0.0570i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (0.466 - 0.884i)T \)
13 \( 1 + (-0.870 - 0.491i)T \)
17 \( 1 + (0.941 + 0.336i)T \)
19 \( 1 + (-0.0285 - 0.999i)T \)
23 \( 1 + (-0.654 + 0.755i)T \)
29 \( 1 + (0.564 - 0.825i)T \)
31 \( 1 + (-0.198 - 0.980i)T \)
37 \( 1 + (-0.736 - 0.676i)T \)
41 \( 1 + (-0.254 + 0.967i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.774 - 0.633i)T \)
53 \( 1 + (0.974 + 0.226i)T \)
59 \( 1 + (0.254 + 0.967i)T \)
61 \( 1 + (-0.998 - 0.0570i)T \)
67 \( 1 + (0.841 + 0.540i)T \)
71 \( 1 + (0.610 + 0.791i)T \)
73 \( 1 + (0.516 + 0.856i)T \)
79 \( 1 + (-0.696 - 0.717i)T \)
83 \( 1 + (0.0855 + 0.996i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
97 \( 1 + (0.466 + 0.884i)T \)
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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

−22.47865326626376606042923583705, −21.54998334586370923385061984852, −21.21848394438813586725559053532, −20.30606181662241654931011897811, −19.39166252560611821126917669417, −18.365059697873823054319060560031, −17.31302419254990503149040706586, −16.54271426533465180940762039388, −15.92395089657894895189464363413, −14.86742281309294666253894342332, −14.37035696650729615682875941211, −13.91994217322275908640346556637, −12.394427730080672712261351104751, −11.95270148915697976333134222312, −10.86763871147281095183079355861, −10.27847532030475306417845236432, −9.59617707978283118605215310871, −8.164729349331996612339940244775, −7.016530166402570871270417484219, −6.28615514580486276729546450374, −5.40638628731254030017411882315, −4.681636223511091195637723214204, −3.56464397204827531101213281541, −2.95278759479296947333515972573, −1.80134378490137452557809333362, 0.8322484659667577776520425396, 1.92909582848377649052915853656, 2.703499559071797957712146133370, 4.05854879279410680154261139032, 5.242407013490817267183083922635, 5.58113596696648703853078363618, 6.59252284720294553000192950976, 7.5436201006269224874528982950, 8.24850737136764180688024051432, 9.61627468209648533375514975713, 10.52757840250111465814244265064, 11.7676837064397748110886476278, 12.11324680921941948211459001045, 13.03818278275244493472114623498, 13.47518027356739657964788480369, 14.318291869250519375397499178411, 15.27337499311999528581138126331, 16.27208380959265407439240474532, 17.105619873776955891623579933356, 17.5641476955676107507967128368, 18.80541044885499221393482904583, 19.79701674380363571383178531169, 20.075980074547215583865151992864, 21.28697469319257866692788788163, 21.7935434402505360498731050661

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