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Properties

Label 2-693-693.373-c0-0-0
Degree $2$
Conductor $693$
Sign $0.723 - 0.690i$
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  + i·2-s + (0.5 − 0.866i)3-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s + i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s − 16-s + (0.866 − 0.5i)17-s + (0.866 − 0.499i)18-s + (−0.866 − 0.5i)19-s + (0.866 − 0.499i)21-s + (−0.866 − 0.5i)22-s + ⋯
L(s)  = 1  + i·2-s + (0.5 − 0.866i)3-s + (0.866 + 0.5i)6-s + (0.866 + 0.5i)7-s + i·8-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (−0.866 − 0.5i)13-s + (−0.5 + 0.866i)14-s − 16-s + (0.866 − 0.5i)17-s + (0.866 − 0.499i)18-s + (−0.866 − 0.5i)19-s + (0.866 − 0.499i)21-s + (−0.866 − 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ 0.723 - 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.225874055\)
\(L(\frac12)\) \(\approx\) \(1.225874055\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 - iT - T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + T + T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 - T + T^{2} \)
61 \( 1 + iT - T^{2} \)
67 \( 1 + T + T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 - iT - T^{2} \)
83 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.80807076072477096046568899450, −9.703091221328674054064600478230, −8.531254437827466964742737693384, −8.060739623734435587848126874244, −7.28691689833962311286164118778, −6.64865596757856175355698790986, −5.48219583505896501398715807099, −4.79309000686873879846392508890, −2.80999549531696596539562086700, −1.97573628137290425263419317540, 1.74362973296127457793905635046, 2.90397894227530734466906787653, 3.85123373120137279118452618740, 4.72393849642230580818059367209, 5.87911316039817196044417703740, 7.34263931032346802991061976180, 8.116150125991887498692709413491, 9.071967513984199956748669926406, 10.04852554188877455906936921779, 10.54568109257295180523296777947

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