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Properties

Label 2-712-712.619-c0-0-0
Degree $2$
Conductor $712$
Sign $0.460 + 0.887i$
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  + (0.841 − 0.540i)2-s + (0.512 − 0.234i)3-s + (0.415 − 0.909i)4-s + (0.304 − 0.474i)6-s + (−0.142 − 0.989i)8-s + (−0.446 + 0.515i)9-s + (−0.0405 + 0.281i)11-s − 0.563i·12-s + (−0.654 − 0.755i)16-s + (−1.41 − 0.909i)17-s + (−0.0971 + 0.675i)18-s + (1.49 + 1.29i)19-s + (0.118 + 0.258i)22-s + (−0.304 − 0.474i)24-s + (−0.959 + 0.281i)25-s + ⋯
L(s)  = 1  + (0.841 − 0.540i)2-s + (0.512 − 0.234i)3-s + (0.415 − 0.909i)4-s + (0.304 − 0.474i)6-s + (−0.142 − 0.989i)8-s + (−0.446 + 0.515i)9-s + (−0.0405 + 0.281i)11-s − 0.563i·12-s + (−0.654 − 0.755i)16-s + (−1.41 − 0.909i)17-s + (−0.0971 + 0.675i)18-s + (1.49 + 1.29i)19-s + (0.118 + 0.258i)22-s + (−0.304 − 0.474i)24-s + (−0.959 + 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.460 + 0.887i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (619, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.460 + 0.887i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.636702544\)
\(L(\frac12)\) \(\approx\) \(1.636702544\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.841 + 0.540i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
good3 \( 1 + (-0.512 + 0.234i)T + (0.654 - 0.755i)T^{2} \)
5 \( 1 + (0.959 - 0.281i)T^{2} \)
7 \( 1 + (-0.959 + 0.281i)T^{2} \)
11 \( 1 + (0.0405 - 0.281i)T + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.654 + 0.755i)T^{2} \)
17 \( 1 + (1.41 + 0.909i)T + (0.415 + 0.909i)T^{2} \)
19 \( 1 + (-1.49 - 1.29i)T + (0.142 + 0.989i)T^{2} \)
23 \( 1 + (-0.142 - 0.989i)T^{2} \)
29 \( 1 + (-0.959 + 0.281i)T^{2} \)
31 \( 1 + (-0.142 + 0.989i)T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.654 + 0.755i)T^{2} \)
43 \( 1 + (1.07 + 0.153i)T + (0.959 + 0.281i)T^{2} \)
47 \( 1 + (0.654 + 0.755i)T^{2} \)
53 \( 1 + (0.654 - 0.755i)T^{2} \)
59 \( 1 + (-0.983 - 0.449i)T + (0.654 + 0.755i)T^{2} \)
61 \( 1 + (0.841 + 0.540i)T^{2} \)
67 \( 1 + (-0.544 - 1.19i)T + (-0.654 + 0.755i)T^{2} \)
71 \( 1 + (0.959 + 0.281i)T^{2} \)
73 \( 1 + (1.25 + 1.45i)T + (-0.142 + 0.989i)T^{2} \)
79 \( 1 + (0.142 - 0.989i)T^{2} \)
83 \( 1 + (-0.983 + 1.53i)T + (-0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
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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

−10.58097194718341895818190253343, −9.754788381812313102086165813255, −8.915289097257024389216746639642, −7.74943314583803145115622001468, −6.97474835542170804432829547976, −5.76973676523139388174947784714, −5.00736924363334770953691511530, −3.84222412218499517445491517628, −2.79952127318780285046360097758, −1.80127156064046227071770881849, 2.40731526164756565277565590906, 3.41065865557710097289770786194, 4.30568402355423778760789285973, 5.39898408811628517191950177534, 6.32588099298392627390247687479, 7.14278088379505225807639267293, 8.231225714979226217181048433817, 8.837141849270743604498494455468, 9.747921118636022691804817930053, 11.11142694888981838109194756730

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