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Properties

Label 2-847-847.223-c0-0-0
Degree $2$
Conductor $847$
Sign $0.0337 - 0.999i$
Analytic cond. $0.422708$
Root an. cond. $0.650160$
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  + (1.07 + 1.11i)2-s + (−0.0398 + 1.39i)4-s + (0.774 − 0.633i)7-s + (−0.453 + 0.415i)8-s + (−0.809 + 0.587i)9-s + (−0.466 + 0.884i)11-s + (1.53 + 0.176i)14-s + (0.444 + 0.0254i)16-s + (−1.52 − 0.264i)18-s + (−1.48 + 0.436i)22-s + (−0.427 + 0.274i)23-s + (0.516 − 0.856i)25-s + (0.853 + 1.10i)28-s + (−0.899 − 1.49i)29-s + (0.853 + 0.985i)32-s + ⋯
L(s)  = 1  + (1.07 + 1.11i)2-s + (−0.0398 + 1.39i)4-s + (0.774 − 0.633i)7-s + (−0.453 + 0.415i)8-s + (−0.809 + 0.587i)9-s + (−0.466 + 0.884i)11-s + (1.53 + 0.176i)14-s + (0.444 + 0.0254i)16-s + (−1.52 − 0.264i)18-s + (−1.48 + 0.436i)22-s + (−0.427 + 0.274i)23-s + (0.516 − 0.856i)25-s + (0.853 + 1.10i)28-s + (−0.899 − 1.49i)29-s + (0.853 + 0.985i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0337 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.0337 - 0.999i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.0337 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696278175\)
\(L(\frac12)\) \(\approx\) \(1.696278175\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.774 + 0.633i)T \)
11 \( 1 + (0.466 - 0.884i)T \)
good2 \( 1 + (-1.07 - 1.11i)T + (-0.0285 + 0.999i)T^{2} \)
3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.516 + 0.856i)T^{2} \)
13 \( 1 + (0.254 + 0.967i)T^{2} \)
17 \( 1 + (0.985 - 0.170i)T^{2} \)
19 \( 1 + (-0.696 - 0.717i)T^{2} \)
23 \( 1 + (0.427 - 0.274i)T + (0.415 - 0.909i)T^{2} \)
29 \( 1 + (0.899 + 1.49i)T + (-0.466 + 0.884i)T^{2} \)
31 \( 1 + (-0.774 + 0.633i)T^{2} \)
37 \( 1 + (0.348 - 0.510i)T + (-0.362 - 0.931i)T^{2} \)
41 \( 1 + (-0.610 + 0.791i)T^{2} \)
43 \( 1 + (0.829 + 1.81i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (-0.941 + 0.336i)T^{2} \)
53 \( 1 + (1.94 - 0.111i)T + (0.993 - 0.113i)T^{2} \)
59 \( 1 + (-0.610 - 0.791i)T^{2} \)
61 \( 1 + (0.0285 + 0.999i)T^{2} \)
67 \( 1 + (0.0243 + 0.169i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (-1.18 + 0.276i)T + (0.897 - 0.441i)T^{2} \)
73 \( 1 + (0.870 - 0.491i)T^{2} \)
79 \( 1 + (-0.374 - 1.84i)T + (-0.921 + 0.389i)T^{2} \)
83 \( 1 + (0.736 - 0.676i)T^{2} \)
89 \( 1 + (0.142 - 0.989i)T^{2} \)
97 \( 1 + (-0.516 - 0.856i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.65161845151352087766809026173, −9.838126971431189295706647234818, −8.385088164664704404486464377476, −7.84909252646799720076560101706, −7.13614046814808509627142812088, −6.16317914890329207938879670922, −5.20340396565370575399907507593, −4.66310700188020992182088649664, −3.67148079771543018630599230941, −2.16488271007868334125917801564, 1.61329669549386469579432364308, 2.87717145629173731193441589856, 3.50220669312087664471634876071, 4.84114887614852012506038455237, 5.45640777672895730783178426653, 6.27056287452594465240549408234, 7.79127449795370358179353195995, 8.632180919004433533801974265455, 9.451664938119252797212333913357, 10.72447738489115351265606035419

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