Properties

Label 2-3-3.2-c10-0-1
Degree $2$
Conductor $3$
Sign $0.111 + 0.993i$
Analytic cond. $1.90607$
Root an. cond. $1.38060$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 26.8i·2-s + (−27 − 241. i)3-s + 303.·4-s + 2.84e3i·5-s + (−6.48e3 + 724. i)6-s + 1.72e4·7-s − 3.56e4i·8-s + (−5.75e4 + 1.30e4i)9-s + 7.63e4·10-s + 1.86e5i·11-s + (−8.20e3 − 7.34e4i)12-s − 1.69e5·13-s − 4.62e5i·14-s + (6.86e5 − 7.67e4i)15-s − 6.44e5·16-s + 3.43e5i·17-s + ⋯
L(s)  = 1  − 0.838i·2-s + (−0.111 − 0.993i)3-s + 0.296·4-s + 0.910i·5-s + (−0.833 + 0.0931i)6-s + 1.02·7-s − 1.08i·8-s + (−0.975 + 0.220i)9-s + 0.763·10-s + 1.15i·11-s + (−0.0329 − 0.295i)12-s − 0.456·13-s − 0.859i·14-s + (0.904 − 0.101i)15-s − 0.614·16-s + 0.241i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.111 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $0.111 + 0.993i$
Analytic conductor: \(1.90607\)
Root analytic conductor: \(1.38060\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :5),\ 0.111 + 0.993i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.999314 - 0.893814i\)
\(L(\frac12)\) \(\approx\) \(0.999314 - 0.893814i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (27 + 241. i)T \)
good2 \( 1 + 26.8iT - 1.02e3T^{2} \)
5 \( 1 - 2.84e3iT - 9.76e6T^{2} \)
7 \( 1 - 1.72e4T + 2.82e8T^{2} \)
11 \( 1 - 1.86e5iT - 2.59e10T^{2} \)
13 \( 1 + 1.69e5T + 1.37e11T^{2} \)
17 \( 1 - 3.43e5iT - 2.01e12T^{2} \)
19 \( 1 + 9.49e5T + 6.13e12T^{2} \)
23 \( 1 + 2.65e6iT - 4.14e13T^{2} \)
29 \( 1 + 3.18e6iT - 4.20e14T^{2} \)
31 \( 1 + 2.97e7T + 8.19e14T^{2} \)
37 \( 1 + 6.08e7T + 4.80e15T^{2} \)
41 \( 1 + 1.81e8iT - 1.34e16T^{2} \)
43 \( 1 - 1.07e8T + 2.16e16T^{2} \)
47 \( 1 - 2.67e8iT - 5.25e16T^{2} \)
53 \( 1 + 1.92e8iT - 1.74e17T^{2} \)
59 \( 1 - 6.49e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.03e9T + 7.13e17T^{2} \)
67 \( 1 - 1.87e9T + 1.82e18T^{2} \)
71 \( 1 + 2.68e9iT - 3.25e18T^{2} \)
73 \( 1 + 2.84e9T + 4.29e18T^{2} \)
79 \( 1 - 1.48e9T + 9.46e18T^{2} \)
83 \( 1 + 1.26e9iT - 1.55e19T^{2} \)
89 \( 1 - 6.02e9iT - 3.11e19T^{2} \)
97 \( 1 + 1.59e9T + 7.37e19T^{2} \)
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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

−24.20318576127480771016158811837, −22.44011139079520987686395056833, −20.58561056656826117159746637880, −19.08117028631012313255617425684, −17.70115337145780505421991111457, −14.67416678267482593441362433697, −12.36693015680932163830038383401, −10.85759080033753171555976318191, −7.16165388174068795820950336271, −2.08462692934217493550687428847, 5.24946800564837304581190225336, 8.499579027723398985905653048973, 11.29941217197126212856117573402, 14.52431261187424724044323929269, 16.11038654296428834026969403329, 17.18075787907553319914893773469, 20.28532374838293855514717649071, 21.47064512361908584399553012875, 23.74338877657051435643787226644, 24.84457899962566353199445389259

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