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Properties

Label 2-847-847.300-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: $\chi_{847} (300, \cdot )$
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

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

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