Properties

Label 2-712-712.99-c0-0-0
Degree $2$
Conductor $712$
Sign $-0.528 - 0.848i$
Analytic cond. $0.355334$
Root an. cond. $0.596099$
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  + (−0.415 + 0.909i)2-s + (−1.94 + 0.139i)3-s + (−0.654 − 0.755i)4-s + (0.682 − 1.83i)6-s + (0.959 − 0.281i)8-s + (2.79 − 0.401i)9-s + (−0.540 − 0.158i)11-s + (1.38 + 1.38i)12-s + (−0.142 + 0.989i)16-s + (−0.755 + 0.345i)17-s + (−0.794 + 2.70i)18-s + (1.40 + 1.05i)19-s + (0.368 − 0.425i)22-s + (−1.83 + 0.682i)24-s + (−0.841 + 0.540i)25-s + ⋯
L(s)  = 1  + (−0.415 + 0.909i)2-s + (−1.94 + 0.139i)3-s + (−0.654 − 0.755i)4-s + (0.682 − 1.83i)6-s + (0.959 − 0.281i)8-s + (2.79 − 0.401i)9-s + (−0.540 − 0.158i)11-s + (1.38 + 1.38i)12-s + (−0.142 + 0.989i)16-s + (−0.755 + 0.345i)17-s + (−0.794 + 2.70i)18-s + (1.40 + 1.05i)19-s + (0.368 − 0.425i)22-s + (−1.83 + 0.682i)24-s + (−0.841 + 0.540i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 - 0.848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $-0.528 - 0.848i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ -0.528 - 0.848i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3262207565\)
\(L(\frac12)\) \(\approx\) \(0.3262207565\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.415 - 0.909i)T \)
89 \( 1 + (0.841 + 0.540i)T \)
good3 \( 1 + (1.94 - 0.139i)T + (0.989 - 0.142i)T^{2} \)
5 \( 1 + (0.841 - 0.540i)T^{2} \)
7 \( 1 + (-0.540 - 0.841i)T^{2} \)
11 \( 1 + (0.540 + 0.158i)T + (0.841 + 0.540i)T^{2} \)
13 \( 1 + (0.989 - 0.142i)T^{2} \)
17 \( 1 + (0.755 - 0.345i)T + (0.654 - 0.755i)T^{2} \)
19 \( 1 + (-1.40 - 1.05i)T + (0.281 + 0.959i)T^{2} \)
23 \( 1 + (0.281 + 0.959i)T^{2} \)
29 \( 1 + (0.540 + 0.841i)T^{2} \)
31 \( 1 + (0.281 - 0.959i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (0.100 - 1.41i)T + (-0.989 - 0.142i)T^{2} \)
43 \( 1 + (-0.898 - 1.64i)T + (-0.540 + 0.841i)T^{2} \)
47 \( 1 + (-0.142 + 0.989i)T^{2} \)
53 \( 1 + (-0.142 - 0.989i)T^{2} \)
59 \( 1 + (-0.697 - 0.0498i)T + (0.989 + 0.142i)T^{2} \)
61 \( 1 + (0.909 - 0.415i)T^{2} \)
67 \( 1 + (-1.29 + 1.49i)T + (-0.142 - 0.989i)T^{2} \)
71 \( 1 + (0.841 + 0.540i)T^{2} \)
73 \( 1 + (0.153 - 1.07i)T + (-0.959 - 0.281i)T^{2} \)
79 \( 1 + (-0.959 - 0.281i)T^{2} \)
83 \( 1 + (0.697 - 1.86i)T + (-0.755 - 0.654i)T^{2} \)
97 \( 1 + (-0.273 + 0.0801i)T + (0.841 - 0.540i)T^{2} \)
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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

−10.95689352607128272411107005354, −9.962773069106447534756336618066, −9.551603650082827202893110056990, −8.022179239277086245051934346768, −7.29028515215613751174928893186, −6.32295256261687292957502672261, −5.72817552213318340738432457170, −5.00716593730445465197913814563, −4.04343792951888424511216687459, −1.28094185708622593531553461456, 0.60152963256656413118709655715, 2.19534363424892827770481020801, 3.96681418739508165666320035133, 4.95471288819158337197923390615, 5.60868631443069227309641886621, 6.97533845391762948453589368613, 7.48484938299490647869404943563, 8.918587054406445712368464916976, 9.931033158754207989073953103991, 10.46734482867067852083148443855

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