Properties

Label 2-322-161.160-c1-0-12
Degree $2$
Conductor $322$
Sign $-0.625 + 0.780i$
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 − 1.41i·3-s + 4-s + 1.41i·6-s − 2.64·7-s − 8-s + 0.999·9-s − 3.74i·11-s − 1.41i·12-s − 4.24i·13-s + 2.64·14-s + 16-s − 0.999·18-s − 5.29·19-s + 3.74i·21-s + 3.74i·22-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.816i·3-s + 0.5·4-s + 0.577i·6-s − 0.999·7-s − 0.353·8-s + 0.333·9-s − 1.12i·11-s − 0.408i·12-s − 1.17i·13-s + 0.707·14-s + 0.250·16-s − 0.235·18-s − 1.21·19-s + 0.816i·21-s + 0.797i·22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.625 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.625 + 0.780i$
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: \((2,\ 322,\ (\ :1/2),\ -0.625 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.296758 - 0.618304i\)
\(L(\frac12)\) \(\approx\) \(0.296758 - 0.618304i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.64T \)
23 \( 1 + (3 - 3.74i)T \)
good3 \( 1 + 1.41iT - 3T^{2} \)
5 \( 1 + 5T^{2} \)
11 \( 1 + 3.74iT - 11T^{2} \)
13 \( 1 + 4.24iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 5.29T + 19T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + 8.48iT - 31T^{2} \)
37 \( 1 + 11.2iT - 37T^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 + 3.74iT - 53T^{2} \)
59 \( 1 - 9.89iT - 59T^{2} \)
61 \( 1 - 10.5T + 61T^{2} \)
67 \( 1 + 11.2iT - 67T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 8.48iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 15.8T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 + 5.29T + 97T^{2} \)
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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

−11.21666698948593266522711476972, −10.24644666603516748781959407450, −9.484801200757308825072762352549, −8.246808524132898051611036485790, −7.65877560421752824588729103866, −6.42358553718720914610377248831, −5.87308308284396725950810075422, −3.76856629766051743070373772046, −2.40418026451990752365784642526, −0.59765340142038398772023628798, 2.10381351862526570217413915343, 3.78545433701755671865717783205, 4.75127959649727462015182895875, 6.46639347526639826809600464490, 7.00860497571241306211756392789, 8.478618341595021166014585116727, 9.336995673578719428254390439549, 10.11328691949484745623730741883, 10.51717120905856676271991472369, 11.96895982996578618770082863728

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