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Properties

Label 2-693-693.340-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: $\chi_{693} (340, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ -0.723 - 0.690i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.102966161\)
\(L(\frac12)\) \(\approx\) \(1.102966161\)
\(L(1)\) 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

−10.95994078732607686889478669743, −10.01645079374216803575093713105, −8.880792064821051560618500263492, −8.613869824760754229613205342935, −7.52903811744511538487449527159, −6.53009785928943558754203728719, −5.64415250114785742628283147997, −4.94315661270667850243468528440, −3.37068613175471927177015663870, −2.65783343258377523508092618454, 1.29490972019142961416170338714, 2.49042934744665076356249358552, 3.36937252403309540669783543008, 4.41520937647298348663243967221, 6.38695152314526729743914500036, 6.64188996599811139169385321715, 7.74440605865458520449370354285, 8.700875914035872939559683694680, 9.798173203390530836039817679402, 10.23343024055184846641457359900

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