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Properties

Label 2-722-19.18-c2-0-13
Degree $2$
Conductor $722$
Sign $-0.397 - 0.917i$
Analytic cond. $19.6730$
Root an. cond. $4.43543$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.41i·2-s + 3.14i·3-s − 2.00·4-s − 5-s + 4.44·6-s + 6.89·7-s + 2.82i·8-s − 0.898·9-s + 1.41i·10-s − 14.8·11-s − 6.29i·12-s + 17.1i·13-s − 9.75i·14-s − 3.14i·15-s + 4.00·16-s − 2.10·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.04i·3-s − 0.500·4-s − 0.200·5-s + 0.741·6-s + 0.985·7-s + 0.353i·8-s − 0.0998·9-s + 0.141i·10-s − 1.35·11-s − 0.524i·12-s + 1.31i·13-s − 0.696i·14-s − 0.209i·15-s + 0.250·16-s − 0.123·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.397 - 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.397 - 0.917i$
Analytic conductor: \(19.6730\)
Root analytic conductor: \(4.43543\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1),\ -0.397 - 0.917i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.129409109\)
\(L(\frac12)\) \(\approx\) \(1.129409109\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
19 \( 1 \)
good3 \( 1 - 3.14iT - 9T^{2} \)
5 \( 1 + T + 25T^{2} \)
7 \( 1 - 6.89T + 49T^{2} \)
11 \( 1 + 14.8T + 121T^{2} \)
13 \( 1 - 17.1iT - 169T^{2} \)
17 \( 1 + 2.10T + 289T^{2} \)
23 \( 1 - 27.0T + 529T^{2} \)
29 \( 1 - 6.40iT - 841T^{2} \)
31 \( 1 + 31.1iT - 961T^{2} \)
37 \( 1 + 28.9iT - 1.36e3T^{2} \)
41 \( 1 - 64.5iT - 1.68e3T^{2} \)
43 \( 1 + 75.3T + 1.84e3T^{2} \)
47 \( 1 + 11.5T + 2.20e3T^{2} \)
53 \( 1 - 80.0iT - 2.80e3T^{2} \)
59 \( 1 - 58.7iT - 3.48e3T^{2} \)
61 \( 1 - 2.19T + 3.72e3T^{2} \)
67 \( 1 - 59.6iT - 4.48e3T^{2} \)
71 \( 1 - 101. iT - 5.04e3T^{2} \)
73 \( 1 + 127.T + 5.32e3T^{2} \)
79 \( 1 + 6.67iT - 6.24e3T^{2} \)
83 \( 1 + 1.30T + 6.88e3T^{2} \)
89 \( 1 + 6.75iT - 7.92e3T^{2} \)
97 \( 1 + 131. iT - 9.40e3T^{2} \)
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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

−10.50013805746492777725388054685, −9.781330913031780818903672186031, −8.991082630186317522462160223919, −8.109033818499829310810766511628, −7.17560873093578800027824378347, −5.58590216623826268514161353615, −4.68157447173436949100346681069, −4.20406447630208634779163294602, −2.90197531166580453849065736813, −1.63494955449855115209803289089, 0.39541794064833733407490269052, 1.81457152382606896090577563121, 3.22019652850234187732438635556, 4.85986332375484930375101997917, 5.39911937220342951507239204231, 6.59138991880090852205105718477, 7.48268138577391442072908949248, 7.994853696079143857798087233695, 8.539664642095292361756340357709, 9.996810101466006522550114757323

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