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Properties

Label 2-712-712.427-c0-0-0
Degree $2$
Conductor $712$
Sign $0.995 + 0.0899i$
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  + (0.654 + 0.755i)2-s + (1.12 − 1.50i)3-s + (−0.142 + 0.989i)4-s + (1.86 − 0.133i)6-s + (−0.841 + 0.540i)8-s + (−0.707 − 2.40i)9-s + (−0.909 − 0.584i)11-s + (1.32 + 1.32i)12-s + (−0.959 − 0.281i)16-s + (0.989 + 0.857i)17-s + (1.35 − 2.11i)18-s + (−0.936 + 1.71i)19-s + (−0.153 − 1.07i)22-s + (−0.133 + 1.86i)24-s + (−0.415 + 0.909i)25-s + ⋯
L(s)  = 1  + (0.654 + 0.755i)2-s + (1.12 − 1.50i)3-s + (−0.142 + 0.989i)4-s + (1.86 − 0.133i)6-s + (−0.841 + 0.540i)8-s + (−0.707 − 2.40i)9-s + (−0.909 − 0.584i)11-s + (1.32 + 1.32i)12-s + (−0.959 − 0.281i)16-s + (0.989 + 0.857i)17-s + (1.35 − 2.11i)18-s + (−0.936 + 1.71i)19-s + (−0.153 − 1.07i)22-s + (−0.133 + 1.86i)24-s + (−0.415 + 0.909i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0899i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.995 + 0.0899i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (427, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.995 + 0.0899i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.660114731\)
\(L(\frac12)\) \(\approx\) \(1.660114731\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (-1.12 + 1.50i)T + (-0.281 - 0.959i)T^{2} \)
5 \( 1 + (0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.909 - 0.415i)T^{2} \)
11 \( 1 + (0.909 + 0.584i)T + (0.415 + 0.909i)T^{2} \)
13 \( 1 + (-0.281 - 0.959i)T^{2} \)
17 \( 1 + (-0.989 - 0.857i)T + (0.142 + 0.989i)T^{2} \)
19 \( 1 + (0.936 - 1.71i)T + (-0.540 - 0.841i)T^{2} \)
23 \( 1 + (-0.540 - 0.841i)T^{2} \)
29 \( 1 + (0.909 + 0.415i)T^{2} \)
31 \( 1 + (-0.540 + 0.841i)T^{2} \)
37 \( 1 + iT^{2} \)
41 \( 1 + (-1.13 + 0.847i)T + (0.281 - 0.959i)T^{2} \)
43 \( 1 + (0.0303 + 0.139i)T + (-0.909 + 0.415i)T^{2} \)
47 \( 1 + (-0.959 - 0.281i)T^{2} \)
53 \( 1 + (-0.959 + 0.281i)T^{2} \)
59 \( 1 + (1.19 + 1.59i)T + (-0.281 + 0.959i)T^{2} \)
61 \( 1 + (0.755 + 0.654i)T^{2} \)
67 \( 1 + (0.0801 + 0.557i)T + (-0.959 + 0.281i)T^{2} \)
71 \( 1 + (0.415 + 0.909i)T^{2} \)
73 \( 1 + (1.74 + 0.512i)T + (0.841 + 0.540i)T^{2} \)
79 \( 1 + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (-1.19 + 0.0855i)T + (0.989 - 0.142i)T^{2} \)
97 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
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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

−10.69627985179855010255984940448, −9.326917101682718658191012927613, −8.351332041406711903547806309320, −7.937076349814035899333734733665, −7.33244846555670625812906500404, −6.17009360647573282158565578171, −5.69189023050477380344924344964, −3.87048865811984084505948243017, −3.09419888256865580070144159930, −1.87780954807291510019875714084, 2.46265067207093483053294352210, 2.93741114013149529147065641854, 4.25794558944904644099003056460, 4.71872531035251595220640034897, 5.67280509704095964558304346322, 7.29368158595115337912888114730, 8.416774182045797212114390479630, 9.270214420550771274287658797681, 9.880143903253433640709461592864, 10.53812239294290037946674538063

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