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Properties

Label 2-712-712.275-c0-0-0
Degree $2$
Conductor $712$
Sign $0.0536 - 0.998i$
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 + (−0.118 + 0.822i)3-s + (−0.142 − 0.989i)4-s + (−0.544 − 0.627i)6-s + (0.841 + 0.540i)8-s + (0.297 + 0.0872i)9-s + (1.41 − 0.909i)11-s + 0.830·12-s + (−0.959 + 0.281i)16-s + (0.857 + 0.989i)17-s + (−0.260 + 0.167i)18-s + (−1.61 − 0.474i)19-s + (−0.239 + 1.66i)22-s + (−0.544 + 0.627i)24-s + (0.415 + 0.909i)25-s + ⋯
L(s)  = 1  + (−0.654 + 0.755i)2-s + (−0.118 + 0.822i)3-s + (−0.142 − 0.989i)4-s + (−0.544 − 0.627i)6-s + (0.841 + 0.540i)8-s + (0.297 + 0.0872i)9-s + (1.41 − 0.909i)11-s + 0.830·12-s + (−0.959 + 0.281i)16-s + (0.857 + 0.989i)17-s + (−0.260 + 0.167i)18-s + (−1.61 − 0.474i)19-s + (−0.239 + 1.66i)22-s + (−0.544 + 0.627i)24-s + (0.415 + 0.909i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0536 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0536 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(712\)    =    \(2^{3} \cdot 89\)
Sign: $0.0536 - 0.998i$
Analytic conductor: \(0.355334\)
Root analytic conductor: \(0.596099\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{712} (275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 712,\ (\ :0),\ 0.0536 - 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7254195044\)
\(L(\frac12)\) \(\approx\) \(0.7254195044\)
\(L(1)\) 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 \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (-0.415 - 0.909i)T^{2} \)
7 \( 1 + (-0.415 - 0.909i)T^{2} \)
11 \( 1 + (-1.41 + 0.909i)T + (0.415 - 0.909i)T^{2} \)
13 \( 1 + (0.959 + 0.281i)T^{2} \)
17 \( 1 + (-0.857 - 0.989i)T + (-0.142 + 0.989i)T^{2} \)
19 \( 1 + (1.61 + 0.474i)T + (0.841 + 0.540i)T^{2} \)
23 \( 1 + (-0.841 - 0.540i)T^{2} \)
29 \( 1 + (-0.415 - 0.909i)T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.284 + 1.97i)T + (-0.959 + 0.281i)T^{2} \)
43 \( 1 + (1.10 - 0.708i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (0.959 - 0.281i)T^{2} \)
53 \( 1 + (0.959 + 0.281i)T^{2} \)
59 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (0.654 + 0.755i)T^{2} \)
67 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
71 \( 1 + (-0.415 + 0.909i)T^{2} \)
73 \( 1 + (0.797 - 0.234i)T + (0.841 - 0.540i)T^{2} \)
79 \( 1 + (-0.841 + 0.540i)T^{2} \)
83 \( 1 + (-0.186 - 0.215i)T + (-0.142 + 0.989i)T^{2} \)
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

−10.67900440654034764105091717669, −9.899304506723388860604676125860, −8.981870210378083536597428825041, −8.551557428574150849634408407759, −7.33795715632991116512121916664, −6.43010668952500930305291606724, −5.65796252106123021614781523064, −4.51673972161283134328017470147, −3.65193831737785199977724193417, −1.54382699945098472470060635177, 1.25713073863435973788773239179, 2.27178452513696149305453461663, 3.75832107948327206604236656655, 4.66707895081450820174290811175, 6.47666619601244325589421793883, 6.94423695627813315619621166495, 7.924427594919711029147097111287, 8.745119111684084102953606619161, 9.763940260494304293722677154986, 10.20661642875140141524849393444

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