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Properties

Label 2-322-7.6-c2-0-0
Degree $2$
Conductor $322$
Sign $-0.372 + 0.928i$
Analytic cond. $8.77386$
Root an. cond. $2.96207$
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  − 1.41·2-s + 5.78i·3-s + 2.00·4-s + 5.81i·5-s − 8.17i·6-s + (2.60 − 6.49i)7-s − 2.82·8-s − 24.4·9-s − 8.21i·10-s − 13.7·11-s + 11.5i·12-s − 20.1i·13-s + (−3.68 + 9.18i)14-s − 33.5·15-s + 4.00·16-s + 13.6i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.92i·3-s + 0.500·4-s + 1.16i·5-s − 1.36i·6-s + (0.372 − 0.928i)7-s − 0.353·8-s − 2.71·9-s − 0.821i·10-s − 1.25·11-s + 0.963i·12-s − 1.55i·13-s + (−0.263 + 0.656i)14-s − 2.23·15-s + 0.250·16-s + 0.801i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.372 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(322\)    =    \(2 \cdot 7 \cdot 23\)
Sign: $-0.372 + 0.928i$
Analytic conductor: \(8.77386\)
Root analytic conductor: \(2.96207\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{322} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 322,\ (\ :1),\ -0.372 + 0.928i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.201789 - 0.298322i\)
\(L(\frac12)\) \(\approx\) \(0.201789 - 0.298322i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
7 \( 1 + (-2.60 + 6.49i)T \)
23 \( 1 + 4.79T \)
good3 \( 1 - 5.78iT - 9T^{2} \)
5 \( 1 - 5.81iT - 25T^{2} \)
11 \( 1 + 13.7T + 121T^{2} \)
13 \( 1 + 20.1iT - 169T^{2} \)
17 \( 1 - 13.6iT - 289T^{2} \)
19 \( 1 - 6.86iT - 361T^{2} \)
29 \( 1 + 36.9T + 841T^{2} \)
31 \( 1 - 26.0iT - 961T^{2} \)
37 \( 1 - 5.03T + 1.36e3T^{2} \)
41 \( 1 - 24.8iT - 1.68e3T^{2} \)
43 \( 1 - 13.2T + 1.84e3T^{2} \)
47 \( 1 - 65.7iT - 2.20e3T^{2} \)
53 \( 1 + 69.6T + 2.80e3T^{2} \)
59 \( 1 + 116. iT - 3.48e3T^{2} \)
61 \( 1 - 52.2iT - 3.72e3T^{2} \)
67 \( 1 + 56.9T + 4.48e3T^{2} \)
71 \( 1 - 65.4T + 5.04e3T^{2} \)
73 \( 1 - 94.2iT - 5.32e3T^{2} \)
79 \( 1 + 37.1T + 6.24e3T^{2} \)
83 \( 1 + 27.7iT - 6.88e3T^{2} \)
89 \( 1 + 53.2iT - 7.92e3T^{2} \)
97 \( 1 - 53.6iT - 9.40e3T^{2} \)
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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

−11.19170733718423463478254687971, −10.73733734398473880043020521110, −10.35484341815383200412627286751, −9.671481698265455187961651625271, −8.267909403722002780609435611602, −7.66039587890154663579143988269, −6.05594549872737464263125995489, −5.03732184363657830767671064452, −3.64563984479275361722057643461, −2.85652596716423298506508436043, 0.20192849383058864001296444507, 1.68081766499105371513342675362, 2.47892946257041919824180169276, 5.09294851250201482672111862870, 5.99739033616685067002018679661, 7.17880804373188424595327753704, 7.903773521971717057177795626851, 8.759341251172409902721921969300, 9.286440933807695875041967135849, 11.19471972165048872248600488677

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