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Properties

Label 2-693-231.167-c0-0-1
Degree $2$
Conductor $693$
Sign $-0.147 - 0.989i$
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  + (−0.610 + 1.87i)2-s + (−2.34 − 1.70i)4-s + (0.587 − 0.809i)7-s + (3.03 − 2.20i)8-s + (0.891 + 0.453i)11-s + (1.16 + 1.59i)14-s + (1.39 + 4.29i)16-s + (−1.39 + 1.39i)22-s + 0.312i·23-s + (0.809 − 0.587i)25-s + (−2.76 + 0.896i)28-s + (−0.734 − 0.533i)29-s − 5.17·32-s + (0.951 + 0.690i)37-s + 0.618i·43-s + (−1.31 − 2.58i)44-s + ⋯
L(s)  = 1  + (−0.610 + 1.87i)2-s + (−2.34 − 1.70i)4-s + (0.587 − 0.809i)7-s + (3.03 − 2.20i)8-s + (0.891 + 0.453i)11-s + (1.16 + 1.59i)14-s + (1.39 + 4.29i)16-s + (−1.39 + 1.39i)22-s + 0.312i·23-s + (0.809 − 0.587i)25-s + (−2.76 + 0.896i)28-s + (−0.734 − 0.533i)29-s − 5.17·32-s + (0.951 + 0.690i)37-s + 0.618i·43-s + (−1.31 − 2.58i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.147 - 0.989i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (629, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ -0.147 - 0.989i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6772496911\)
\(L(\frac12)\) \(\approx\) \(0.6772496911\)
\(L(1)\) 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 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (-0.891 - 0.453i)T \)
good2 \( 1 + (0.610 - 1.87i)T + (-0.809 - 0.587i)T^{2} \)
5 \( 1 + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.809 - 0.587i)T^{2} \)
17 \( 1 + (0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.309 - 0.951i)T^{2} \)
23 \( 1 - 0.312iT - T^{2} \)
29 \( 1 + (0.734 + 0.533i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.951 - 0.690i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (-0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + (0.309 - 0.951i)T^{2} \)
53 \( 1 + (-0.863 - 0.280i)T + (0.809 + 0.587i)T^{2} \)
59 \( 1 + (0.309 + 0.951i)T^{2} \)
61 \( 1 + (-0.809 + 0.587i)T^{2} \)
67 \( 1 - 0.618T + T^{2} \)
71 \( 1 + (1.69 - 0.550i)T + (0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.309 + 0.951i)T^{2} \)
79 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.809 + 0.587i)T^{2} \)
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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

−10.45936253091187951372935840207, −9.757995265342043893447273571597, −8.948127302568418380120890984609, −8.122350830311265845999788676471, −7.37085698485431666545576085081, −6.71816112347305667068705749878, −5.82375140160923125137920304210, −4.71336327498026176359474484415, −4.07489034985111058597120614432, −1.26000879937970853575463934527, 1.35212972536604054835604403879, 2.47461377835267084633176270288, 3.53562106782954021963187066022, 4.53278487157347347989921234145, 5.60046733470961286680597776639, 7.31956426218587169612644952584, 8.436389323149689825361390184033, 8.922204059075404120814799321401, 9.592240195721238153529112208441, 10.64726863279264593003175835788

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