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Properties

Label 2-693-693.142-c0-0-0
Degree $2$
Conductor $693$
Sign $-0.805 - 0.592i$
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  + (−0.866 + 0.5i)2-s − 3-s + (0.866 − 0.5i)6-s + i·7-s − i·8-s + 9-s + 11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.866 + 0.5i)19-s − i·21-s + (−0.866 + 0.5i)22-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s − 3-s + (0.866 − 0.5i)6-s + i·7-s − i·8-s + 9-s + 11-s + (−0.866 + 0.5i)13-s + (−0.5 − 0.866i)14-s + (0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.866 + 0.5i)19-s − i·21-s + (−0.866 + 0.5i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 - 0.592i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $-0.805 - 0.592i$
Analytic conductor: \(0.345852\)
Root analytic conductor: \(0.588091\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :0),\ -0.805 - 0.592i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3290186810\)
\(L(\frac12)\) \(\approx\) \(0.3290186810\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
7 \( 1 - iT \)
11 \( 1 - T \)
good2 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + T^{2} \)
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 \( 1 + T^{2} \)
29 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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

−11.02059289622235622184081572064, −9.709131168396113793646659566776, −9.492228642594250159344683578204, −8.465910196334360754818886064538, −7.48349076438773609214535094581, −6.64937005529549119439292382076, −5.95278519445631389242595311095, −4.79662174734155200966589507827, −3.70003429576952060167043918326, −1.72623772032619290438570207430, 0.55864302332698680589390271186, 1.95555743175360643585909157277, 3.83878022620852692052441144924, 4.90954648323117538713824475601, 5.77720704415723209471935105765, 7.07393388190383965141946369460, 7.53156842560459565478828000769, 8.983887958016493938173651549181, 9.666126774225066779772812679028, 10.29258685618434192186331977591

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