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Properties

Label 2-693-231.230-c0-0-0
Degree $2$
Conductor $693$
Sign $0.169 - 0.985i$
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  − 1.41·2-s + 1.00·4-s + i·7-s + (−0.707 + 0.707i)11-s − 1.41i·14-s − 0.999·16-s + (1.00 − 1.00i)22-s + 1.41i·23-s − 25-s + 1.00i·28-s + 1.41·29-s + 1.41·32-s + 2i·43-s + (−0.707 + 0.707i)44-s − 2.00i·46-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.00·4-s + i·7-s + (−0.707 + 0.707i)11-s − 1.41i·14-s − 0.999·16-s + (1.00 − 1.00i)22-s + 1.41i·23-s − 25-s + 1.00i·28-s + 1.41·29-s + 1.41·32-s + 2i·43-s + (−0.707 + 0.707i)44-s − 2.00i·46-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (692, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3993282220\)
\(L(\frac12)\) \(\approx\) \(0.3993282220\)
\(L(1)\) 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 \( 1 - iT \)
11 \( 1 + (0.707 - 0.707i)T \)
good2 \( 1 + 1.41T + T^{2} \)
5 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - 1.41iT - T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 2iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - 1.41iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 + 1.41iT - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−10.62572838541890158139887712814, −9.676584719347191205127290954832, −9.367187546336412953914951633609, −8.218978160547024675484526126859, −7.79785749450174846199385830982, −6.72991067993121959940788052467, −5.62219871639324428761235871891, −4.54618691855798023683318831545, −2.81745066199515861802531291347, −1.69081418537146588775779779606, 0.71140995498141051228411297980, 2.34117007025028616547990492840, 3.85222700926870261178291696546, 5.03128552597349329113873348737, 6.44095349435658900833444183594, 7.21442380919218726231741030513, 8.172069181875943266583177641645, 8.560140653364413495650812943524, 9.761863802614826319558208315079, 10.35604468254015583719399763172

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