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Properties

Label 2-693-77.69-c0-0-2
Degree $2$
Conductor $693$
Sign $-0.0237 + 0.999i$
Analytic cond. $0.345852$
Root an. cond. $0.588091$
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  + (1.53 − 1.11i)2-s + (0.809 − 2.48i)4-s + (−0.309 + 0.951i)7-s + (−0.951 − 2.92i)8-s + (−0.587 + 0.809i)11-s + (0.587 + 1.80i)14-s + (−2.61 − 1.90i)16-s + 1.90i·22-s − 1.90·23-s + (0.309 + 0.951i)25-s + (2.11 + 1.53i)28-s + (0.363 − 1.11i)29-s − 3.07·32-s + (−0.190 + 0.587i)37-s + 1.61·43-s + (1.53 + 2.11i)44-s + ⋯
L(s)  = 1  + (1.53 − 1.11i)2-s + (0.809 − 2.48i)4-s + (−0.309 + 0.951i)7-s + (−0.951 − 2.92i)8-s + (−0.587 + 0.809i)11-s + (0.587 + 1.80i)14-s + (−2.61 − 1.90i)16-s + 1.90i·22-s − 1.90·23-s + (0.309 + 0.951i)25-s + (2.11 + 1.53i)28-s + (0.363 − 1.11i)29-s − 3.07·32-s + (−0.190 + 0.587i)37-s + 1.61·43-s + (1.53 + 2.11i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.0237 + 0.999i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (685, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ -0.0237 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.932413979\)
\(L(\frac12)\) \(\approx\) \(1.932413979\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 + (0.587 - 0.809i)T \)
good2 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
5 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (-0.309 - 0.951i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + 1.90T + T^{2} \)
29 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.951 + 0.690i)T + (0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + 1.61T + T^{2} \)
71 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−10.56812824736234657482184728762, −9.986086672877696426946873034380, −9.143294650652830672039370834483, −7.69637260956367440873120483912, −6.37428172056037383727966413749, −5.70369387480298725218689283620, −4.82892934506862879160425075063, −3.88266277948631341195482611912, −2.72843968232655013146566319718, −1.94222105362104834733270621172, 2.70559337289646791062308374238, 3.76844763296416656632883612637, 4.47432701505470193106817488578, 5.62512215873889630024047237436, 6.26527731961912200346848472135, 7.21359993919982870612555439628, 7.88621813117907573768493540191, 8.732942047037164419603714378055, 10.25289305499385228751874090149, 11.04684445354138393598332784555

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