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Properties

Label 2-693-11.5-c1-0-19
Degree $2$
Conductor $693$
Sign $0.773 + 0.633i$
Analytic cond. $5.53363$
Root an. cond. $2.35236$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0347 + 0.106i)2-s + (1.60 − 1.16i)4-s + (−0.327 + 1.00i)5-s + (0.809 − 0.587i)7-s + (0.362 + 0.263i)8-s − 0.119·10-s + (1.98 − 2.65i)11-s + (−1.80 − 5.55i)13-s + (0.0909 + 0.0661i)14-s + (1.21 − 3.73i)16-s + (−0.866 + 2.66i)17-s + (2.88 + 2.09i)19-s + (0.651 + 2.00i)20-s + (0.353 + 0.119i)22-s − 4.72·23-s + ⋯
L(s)  = 1  + (0.0245 + 0.0756i)2-s + (0.803 − 0.584i)4-s + (−0.146 + 0.451i)5-s + (0.305 − 0.222i)7-s + (0.128 + 0.0932i)8-s − 0.0377·10-s + (0.597 − 0.801i)11-s + (−0.500 − 1.54i)13-s + (0.0243 + 0.0176i)14-s + (0.303 − 0.933i)16-s + (−0.210 + 0.647i)17-s + (0.661 + 0.480i)19-s + (0.145 + 0.448i)20-s + (0.0753 + 0.0255i)22-s − 0.984·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.773 + 0.633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(693\)    =    \(3^{2} \cdot 7 \cdot 11\)
Sign: $0.773 + 0.633i$
Analytic conductor: \(5.53363\)
Root analytic conductor: \(2.35236\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{693} (379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 693,\ (\ :1/2),\ 0.773 + 0.633i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76251 - 0.629781i\)
\(L(\frac12)\) \(\approx\) \(1.76251 - 0.629781i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (-1.98 + 2.65i)T \)
good2 \( 1 + (-0.0347 - 0.106i)T + (-1.61 + 1.17i)T^{2} \)
5 \( 1 + (0.327 - 1.00i)T + (-4.04 - 2.93i)T^{2} \)
13 \( 1 + (1.80 + 5.55i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (0.866 - 2.66i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.88 - 2.09i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.72T + 23T^{2} \)
29 \( 1 + (-6.95 + 5.05i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.538 - 1.65i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (3.83 - 2.78i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.96 - 3.60i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 5.25T + 43T^{2} \)
47 \( 1 + (6.62 + 4.81i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.43 + 13.6i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-3.15 + 2.29i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (4.55 - 14.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 8.29T + 67T^{2} \)
71 \( 1 + (3.59 - 11.0i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.39 + 2.46i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.10 - 6.47i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.360 - 1.10i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + 2.90T + 89T^{2} \)
97 \( 1 + (1.49 + 4.61i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.32716571556028276345041836846, −9.900593959953685812256208183339, −8.397010622058119124896394222758, −7.76417835132439392785631265361, −6.74234260257564394437068948208, −5.97959576410244989336458022992, −5.11724267241419423953259838057, −3.62437368967948722378167222983, −2.61227492230740848454185818757, −1.08049585201387747168903951982, 1.65519503207097718040563593087, 2.71736696345648464585247407988, 4.16685752736285406056016357543, 4.85748337532448608420717083763, 6.37612962014040621678757963041, 7.04692391603075096979854313028, 7.83604677697184387788742663350, 8.933651145771832707155152137094, 9.515896550687048706142830931584, 10.73972347586883073570192077899

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