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Properties

Label 2-847-77.27-c0-0-2
Degree $2$
Conductor $847$
Sign $0.353 + 0.935i$
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  + (−0.190 − 0.587i)2-s + (0.5 − 0.363i)4-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.5 − 0.363i)14-s + (0.5 − 0.363i)18-s + 0.618·23-s + (−0.809 − 0.587i)25-s + (0.190 − 0.587i)28-s + (−1.30 + 0.951i)29-s − 32-s + (0.5 + 0.363i)36-s + (−0.5 + 0.363i)37-s + 1.61·43-s + ⋯
L(s)  = 1  + (−0.190 − 0.587i)2-s + (0.5 − 0.363i)4-s + (0.809 − 0.587i)7-s + (−0.809 − 0.587i)8-s + (0.309 + 0.951i)9-s + (−0.5 − 0.363i)14-s + (0.5 − 0.363i)18-s + 0.618·23-s + (−0.809 − 0.587i)25-s + (0.190 − 0.587i)28-s + (−1.30 + 0.951i)29-s − 32-s + (0.5 + 0.363i)36-s + (−0.5 + 0.363i)37-s + 1.61·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 + 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.353 + 0.935i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.353 + 0.935i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.083956412\)
\(L(\frac12)\) \(\approx\) \(1.083956412\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (0.809 + 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 + 0.587i)T^{2} \)
19 \( 1 + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - 0.618T + T^{2} \)
29 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
59 \( 1 + (-0.309 + 0.951i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (-0.5 - 1.53i)T + (-0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−10.42169998004485748002979713577, −9.656869425437965260355849490587, −8.604373273599593863654901846555, −7.59794822797662234955189749411, −6.99395553675273423287851391680, −5.76452209852318954133288056498, −4.87290760744827597216012785687, −3.73840730409916363445014737879, −2.36892351673510165607357925606, −1.42357046905662104127875617231, 1.85336112710442420659876314806, 3.11228555477070116113764548662, 4.27722434300912710149458402584, 5.62320536786904700001922524449, 6.18144069533453227873804332488, 7.36245385184377265432474173421, 7.76454873666893867439025673086, 8.955255983075505801835810890773, 9.292776361548392863095321906644, 10.68598731605459716481677943968

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