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Properties

Label 2-847-847.48-c0-0-0
Degree $2$
Conductor $847$
Sign $0.914 - 0.403i$
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  + (−1.56 + 0.882i)2-s + (1.14 − 1.90i)4-s + (0.897 − 0.441i)7-s + (−0.0625 + 2.19i)8-s + (−0.809 − 0.587i)9-s + (0.198 + 0.980i)11-s + (−1.01 + 1.48i)14-s + (−0.799 − 1.51i)16-s + (1.78 + 0.204i)18-s + (−1.17 − 1.35i)22-s + (0.280 − 1.95i)23-s + (0.774 − 0.633i)25-s + (0.190 − 2.21i)28-s + (1.45 + 1.19i)29-s + (0.742 + 0.476i)32-s + ⋯
L(s)  = 1  + (−1.56 + 0.882i)2-s + (1.14 − 1.90i)4-s + (0.897 − 0.441i)7-s + (−0.0625 + 2.19i)8-s + (−0.809 − 0.587i)9-s + (0.198 + 0.980i)11-s + (−1.01 + 1.48i)14-s + (−0.799 − 1.51i)16-s + (1.78 + 0.204i)18-s + (−1.17 − 1.35i)22-s + (0.280 − 1.95i)23-s + (0.774 − 0.633i)25-s + (0.190 − 2.21i)28-s + (1.45 + 1.19i)29-s + (0.742 + 0.476i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 - 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.914 - 0.403i$
Analytic conductor: \(0.422708\)
Root analytic conductor: \(0.650160\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 847,\ (\ :0),\ 0.914 - 0.403i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5034789287\)
\(L(\frac12)\) \(\approx\) \(0.5034789287\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.897 + 0.441i)T \)
11 \( 1 + (-0.198 - 0.980i)T \)
good2 \( 1 + (1.56 - 0.882i)T + (0.516 - 0.856i)T^{2} \)
3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (-0.774 + 0.633i)T^{2} \)
13 \( 1 + (0.985 - 0.170i)T^{2} \)
17 \( 1 + (-0.993 + 0.113i)T^{2} \)
19 \( 1 + (0.870 - 0.491i)T^{2} \)
23 \( 1 + (-0.280 + 1.95i)T + (-0.959 - 0.281i)T^{2} \)
29 \( 1 + (-1.45 - 1.19i)T + (0.198 + 0.980i)T^{2} \)
31 \( 1 + (-0.897 + 0.441i)T^{2} \)
37 \( 1 + (0.569 + 0.240i)T + (0.696 + 0.717i)T^{2} \)
41 \( 1 + (-0.0855 - 0.996i)T^{2} \)
43 \( 1 + (-0.895 + 0.262i)T + (0.841 - 0.540i)T^{2} \)
47 \( 1 + (-0.974 + 0.226i)T^{2} \)
53 \( 1 + (-0.338 + 0.641i)T + (-0.564 - 0.825i)T^{2} \)
59 \( 1 + (-0.0855 + 0.996i)T^{2} \)
61 \( 1 + (-0.516 - 0.856i)T^{2} \)
67 \( 1 + (0.829 - 1.81i)T + (-0.654 - 0.755i)T^{2} \)
71 \( 1 + (0.0620 - 0.159i)T + (-0.736 - 0.676i)T^{2} \)
73 \( 1 + (-0.941 + 0.336i)T^{2} \)
79 \( 1 + (-1.18 - 1.54i)T + (-0.254 + 0.967i)T^{2} \)
83 \( 1 + (0.0285 - 0.999i)T^{2} \)
89 \( 1 + (-0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.774 - 0.633i)T^{2} \)
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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

−10.46874848808380904390015389177, −9.354193005520817450856984706280, −8.572770436461026363904497239245, −8.201427775510295442955859847598, −6.95333667784314977845930233742, −6.70039870269663817642871297258, −5.43565008531586587670693974497, −4.41795286223806403484934551198, −2.49254413822245173933837881851, −1.03699163043897624869914898381, 1.28947261076670231649902458335, 2.49318329267615378582291260029, 3.39404504714056911796891050700, 5.03025920316182054748453053456, 6.09811907928549708494033232805, 7.52445572758515561686628028633, 8.050827641763656157352337794836, 8.818823082229829331394885817127, 9.312114746563803880322485874587, 10.48390274744416075637068461569

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