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Properties

Label 2-847-847.335-c0-0-0
Degree $2$
Conductor $847$
Sign $-0.750 - 0.660i$
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  + (−0.495 + 1.88i)2-s + (−2.43 − 1.37i)4-s + (0.974 + 0.226i)7-s + (2.43 − 2.50i)8-s + (0.309 + 0.951i)9-s + (0.774 − 0.633i)11-s + (−0.909 + 1.72i)14-s + (2.07 + 3.43i)16-s + (−1.94 + 0.111i)18-s + (0.809 + 1.77i)22-s + (−0.112 + 0.129i)23-s + (0.941 + 0.336i)25-s + (−2.05 − 1.88i)28-s + (−1.85 + 0.662i)29-s + (−4.15 + 1.21i)32-s + ⋯
L(s)  = 1  + (−0.495 + 1.88i)2-s + (−2.43 − 1.37i)4-s + (0.974 + 0.226i)7-s + (2.43 − 2.50i)8-s + (0.309 + 0.951i)9-s + (0.774 − 0.633i)11-s + (−0.909 + 1.72i)14-s + (2.07 + 3.43i)16-s + (−1.94 + 0.111i)18-s + (0.809 + 1.77i)22-s + (−0.112 + 0.129i)23-s + (0.941 + 0.336i)25-s + (−2.05 − 1.88i)28-s + (−1.85 + 0.662i)29-s + (−4.15 + 1.21i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.750 - 0.660i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.750 - 0.660i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (335, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ -0.750 - 0.660i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7696496777\)
\(L(\frac12)\) \(\approx\) \(0.7696496777\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.974 - 0.226i)T \)
11 \( 1 + (-0.774 + 0.633i)T \)
good2 \( 1 + (0.495 - 1.88i)T + (-0.870 - 0.491i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.941 - 0.336i)T^{2} \)
13 \( 1 + (-0.0855 + 0.996i)T^{2} \)
17 \( 1 + (0.998 + 0.0570i)T^{2} \)
19 \( 1 + (0.254 - 0.967i)T^{2} \)
23 \( 1 + (0.112 - 0.129i)T + (-0.142 - 0.989i)T^{2} \)
29 \( 1 + (1.85 - 0.662i)T + (0.774 - 0.633i)T^{2} \)
31 \( 1 + (-0.974 - 0.226i)T^{2} \)
37 \( 1 + (0.321 + 1.58i)T + (-0.921 + 0.389i)T^{2} \)
41 \( 1 + (0.736 - 0.676i)T^{2} \)
43 \( 1 + (0.146 - 1.02i)T + (-0.959 - 0.281i)T^{2} \)
47 \( 1 + (-0.993 - 0.113i)T^{2} \)
53 \( 1 + (0.582 - 0.966i)T + (-0.466 - 0.884i)T^{2} \)
59 \( 1 + (0.736 + 0.676i)T^{2} \)
61 \( 1 + (0.870 - 0.491i)T^{2} \)
67 \( 1 + (0.0480 + 0.0308i)T + (0.415 + 0.909i)T^{2} \)
71 \( 1 + (-0.831 + 1.21i)T + (-0.362 - 0.931i)T^{2} \)
73 \( 1 + (0.985 + 0.170i)T^{2} \)
79 \( 1 + (-1.78 + 0.876i)T + (0.610 - 0.791i)T^{2} \)
83 \( 1 + (-0.696 + 0.717i)T^{2} \)
89 \( 1 + (-0.841 + 0.540i)T^{2} \)
97 \( 1 + (-0.941 + 0.336i)T^{2} \)
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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

−10.60218722134249224754291036881, −9.290046456205446564643181654580, −8.930606270447728735214081539807, −7.904229406514243931090493463342, −7.51190803904067399213897231936, −6.51936001887496449827670455370, −5.54022098237817787944845790049, −4.96899954736417721759768178109, −3.94675994725805613508948659945, −1.53022807394440578837940603267, 1.17614055958952652708138995090, 2.14625501668189994936332227062, 3.55111954222754384229023687480, 4.21735669215964984648523622226, 5.13634718651200048598736175571, 6.85210108459415571316554522588, 7.936408510377426571332395339251, 8.748798339930315270313616760710, 9.480821998622384066916780530137, 10.09428199196487827976225785357

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