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Properties

Label 2-712-712.339-c0-0-0
Degree $2$
Conductor $712$
Sign $0.946 + 0.322i$
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  + (0.959 + 0.281i)2-s + (0.254 − 1.17i)3-s + (0.841 + 0.540i)4-s + (0.574 − 1.05i)6-s + (0.654 + 0.755i)8-s + (−0.397 − 0.181i)9-s + (−0.989 + 1.14i)11-s + (0.847 − 0.847i)12-s + (0.415 + 0.909i)16-s + (−0.540 − 1.84i)17-s + (−0.329 − 0.285i)18-s + (−1.86 + 0.697i)19-s + (−1.27 + 0.817i)22-s + (1.05 − 0.574i)24-s + (0.142 − 0.989i)25-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)2-s + (0.254 − 1.17i)3-s + (0.841 + 0.540i)4-s + (0.574 − 1.05i)6-s + (0.654 + 0.755i)8-s + (−0.397 − 0.181i)9-s + (−0.989 + 1.14i)11-s + (0.847 − 0.847i)12-s + (0.415 + 0.909i)16-s + (−0.540 − 1.84i)17-s + (−0.329 − 0.285i)18-s + (−1.86 + 0.697i)19-s + (−1.27 + 0.817i)22-s + (1.05 − 0.574i)24-s + (0.142 − 0.989i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.946 + 0.322i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.946 + 0.322i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.696915358\)
\(L(\frac12)\) \(\approx\) \(1.696915358\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (-0.142 - 0.989i)T \)
good3 \( 1 + (-0.254 + 1.17i)T + (-0.909 - 0.415i)T^{2} \)
5 \( 1 + (-0.142 + 0.989i)T^{2} \)
7 \( 1 + (-0.989 - 0.142i)T^{2} \)
11 \( 1 + (0.989 - 1.14i)T + (-0.142 - 0.989i)T^{2} \)
13 \( 1 + (-0.909 - 0.415i)T^{2} \)
17 \( 1 + (0.540 + 1.84i)T + (-0.841 + 0.540i)T^{2} \)
19 \( 1 + (1.86 - 0.697i)T + (0.755 - 0.654i)T^{2} \)
23 \( 1 + (0.755 - 0.654i)T^{2} \)
29 \( 1 + (0.989 + 0.142i)T^{2} \)
31 \( 1 + (0.755 + 0.654i)T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + (1.38 - 0.300i)T + (0.909 - 0.415i)T^{2} \)
43 \( 1 + (-0.125 - 1.75i)T + (-0.989 + 0.142i)T^{2} \)
47 \( 1 + (0.415 + 0.909i)T^{2} \)
53 \( 1 + (0.415 - 0.909i)T^{2} \)
59 \( 1 + (0.203 + 0.936i)T + (-0.909 + 0.415i)T^{2} \)
61 \( 1 + (-0.281 - 0.959i)T^{2} \)
67 \( 1 + (-1.53 + 0.983i)T + (0.415 - 0.909i)T^{2} \)
71 \( 1 + (-0.142 - 0.989i)T^{2} \)
73 \( 1 + (-0.822 - 1.80i)T + (-0.654 + 0.755i)T^{2} \)
79 \( 1 + (-0.654 + 0.755i)T^{2} \)
83 \( 1 + (-0.203 + 0.373i)T + (-0.540 - 0.841i)T^{2} \)
97 \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \)
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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

−10.78919295271256535931573505341, −9.843218574066555224681333822679, −8.373512629122390398940370595914, −7.79219440984078090445450028402, −6.91316061963987282024865093680, −6.43793640065834679523765318651, −5.08310310578994864551528387700, −4.34024990115418796834812622762, −2.67780527048255904571806746803, −2.04731181142751525673010614761, 2.19075390156466861190449203799, 3.45187047598937766568328294992, 4.08913216992917393358716003836, 5.07999708848955359241382965514, 5.92907607133744017287346231013, 6.90658957361338669484859591980, 8.339827717729325999340384066188, 8.963715613712046270944240705483, 10.40593322802782063649443143649, 10.53874313191497320438061817596

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