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Properties

Label 2-322-161.160-c1-0-4
Degree $2$
Conductor $322$
Sign $0.571 - 0.820i$
Analytic cond. $2.57118$
Root an. cond. $1.60349$
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  + 2-s + 0.936i·3-s + 4-s − 2·5-s + 0.936i·6-s + (1.56 + 2.13i)7-s + 8-s + 2.12·9-s − 2·10-s + 0.936i·11-s + 0.936i·12-s + 3.33i·13-s + (1.56 + 2.13i)14-s − 1.87i·15-s + 16-s − 1.12·17-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.540i·3-s + 0.5·4-s − 0.894·5-s + 0.382i·6-s + (0.590 + 0.807i)7-s + 0.353·8-s + 0.707·9-s − 0.632·10-s + 0.282i·11-s + 0.270i·12-s + 0.924i·13-s + (0.417 + 0.570i)14-s − 0.483i·15-s + 0.250·16-s − 0.272·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.571 - 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $0.571 - 0.820i$
Analytic conductor: \(2.57118\)
Root analytic conductor: \(1.60349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1/2),\ 0.571 - 0.820i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.66698 + 0.871005i\)
\(L(\frac12)\) \(\approx\) \(1.66698 + 0.871005i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
7 \( 1 + (-1.56 - 2.13i)T \)
23 \( 1 + (1.56 + 4.53i)T \)
good3 \( 1 - 0.936iT - 3T^{2} \)
5 \( 1 + 2T + 5T^{2} \)
11 \( 1 - 0.936iT - 11T^{2} \)
13 \( 1 - 3.33iT - 13T^{2} \)
17 \( 1 + 1.12T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + 9.06iT - 31T^{2} \)
37 \( 1 + 3.33iT - 37T^{2} \)
41 \( 1 + 6.67iT - 41T^{2} \)
43 \( 1 - 0.936iT - 43T^{2} \)
47 \( 1 + 6.14iT - 47T^{2} \)
53 \( 1 + 0.410iT - 53T^{2} \)
59 \( 1 + 2.80iT - 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 - 7.60iT - 67T^{2} \)
71 \( 1 + 9.36T + 71T^{2} \)
73 \( 1 - 3.74iT - 73T^{2} \)
79 \( 1 + 12.8iT - 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 2T + 89T^{2} \)
97 \( 1 + 12.2T + 97T^{2} \)
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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

−11.75502336751366417543982655757, −11.17792869564998267191505706892, −9.981447381318622823729636192815, −8.987593026621618707291035913282, −7.83830983452160466298215998408, −6.94981517359820893121793937667, −5.58898548280695186625025921196, −4.50088945490560284169222347052, −3.85104856109090442555194094769, −2.14993086978748661034891363599, 1.30630764907951352205816814227, 3.27743233855085841537162551341, 4.27874832621618955336960424432, 5.37346300531111897966255538084, 6.79455537371617135669563497986, 7.59181757350155572662655794130, 8.145937690190340947655480001157, 9.849682875845921837089508985893, 10.84503668050004291359699294554, 11.62839258509230691004998554537

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