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Properties

Label 2-847-847.489-c0-0-0
Degree $2$
Conductor $847$
Sign $0.338 + 0.940i$
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  + (0.630 − 1.04i)2-s + (−0.229 − 0.434i)4-s + (0.610 − 0.791i)7-s + (0.619 + 0.0354i)8-s + (0.309 + 0.951i)9-s + (−0.921 + 0.389i)11-s + (−0.443 − 1.13i)14-s + (0.705 − 1.03i)16-s + (1.18 + 0.276i)18-s + (−0.173 + 1.20i)22-s + (−1.80 − 0.530i)23-s + (0.198 − 0.980i)25-s + (−0.484 − 0.0838i)28-s + (0.307 + 1.51i)29-s + (−0.376 − 0.823i)32-s + ⋯
L(s)  = 1  + (0.630 − 1.04i)2-s + (−0.229 − 0.434i)4-s + (0.610 − 0.791i)7-s + (0.619 + 0.0354i)8-s + (0.309 + 0.951i)9-s + (−0.921 + 0.389i)11-s + (−0.443 − 1.13i)14-s + (0.705 − 1.03i)16-s + (1.18 + 0.276i)18-s + (−0.173 + 1.20i)22-s + (−1.80 − 0.530i)23-s + (0.198 − 0.980i)25-s + (−0.484 − 0.0838i)28-s + (0.307 + 1.51i)29-s + (−0.376 − 0.823i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.338 + 0.940i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (489, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.338 + 0.940i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.467485385\)
\(L(\frac12)\) \(\approx\) \(1.467485385\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.610 + 0.791i)T \)
11 \( 1 + (0.921 - 0.389i)T \)
good2 \( 1 + (-0.630 + 1.04i)T + (-0.466 - 0.884i)T^{2} \)
3 \( 1 + (-0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.198 + 0.980i)T^{2} \)
13 \( 1 + (-0.941 + 0.336i)T^{2} \)
17 \( 1 + (-0.974 + 0.226i)T^{2} \)
19 \( 1 + (-0.516 + 0.856i)T^{2} \)
23 \( 1 + (1.80 + 0.530i)T + (0.841 + 0.540i)T^{2} \)
29 \( 1 + (-0.307 - 1.51i)T + (-0.921 + 0.389i)T^{2} \)
31 \( 1 + (-0.610 + 0.791i)T^{2} \)
37 \( 1 + (1.12 + 1.16i)T + (-0.0285 + 0.999i)T^{2} \)
41 \( 1 + (0.985 - 0.170i)T^{2} \)
43 \( 1 + (0.949 - 0.610i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.897 + 0.441i)T^{2} \)
53 \( 1 + (-0.831 - 1.21i)T + (-0.362 + 0.931i)T^{2} \)
59 \( 1 + (0.985 + 0.170i)T^{2} \)
61 \( 1 + (0.466 - 0.884i)T^{2} \)
67 \( 1 + (1.30 + 1.50i)T + (-0.142 + 0.989i)T^{2} \)
71 \( 1 + (-1.45 - 1.33i)T + (0.0855 + 0.996i)T^{2} \)
73 \( 1 + (-0.774 + 0.633i)T^{2} \)
79 \( 1 + (0.456 - 1.73i)T + (-0.870 - 0.491i)T^{2} \)
83 \( 1 + (0.998 + 0.0570i)T^{2} \)
89 \( 1 + (0.654 - 0.755i)T^{2} \)
97 \( 1 + (-0.198 - 0.980i)T^{2} \)
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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

−10.59575763518287535293848930696, −9.938733739988874143598620366841, −8.349732072010044431926338597948, −7.75724028512602297027700966183, −6.93997221254381823378625619604, −5.36256662282376575724064436578, −4.64385473321824640591739169053, −3.90276767362290950930654575108, −2.57191939905909935259409641512, −1.69122283636720265991462412379, 1.87562689527910052394724643331, 3.45440767357166424760551563376, 4.56276438966555318198679959973, 5.50167498124512416723961895110, 6.03860700910154455020437790097, 6.99768619683989863754296837173, 7.946178513853938531980195704242, 8.509681496862185327105531317174, 9.714800047845071491786562408299, 10.44988131569587611323435841590

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