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Properties

Label 2-3-3.2-c14-0-3
Degree $2$
Conductor $3$
Sign $-0.879 + 0.476i$
Analytic cond. $3.72986$
Root an. cond. $1.93128$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 225. i·2-s + (1.92e3 − 1.04e3i)3-s − 3.45e4·4-s + 2.31e4i·5-s + (−2.35e5 − 4.33e5i)6-s + 4.78e5·7-s + 4.09e6i·8-s + (2.61e6 − 4.00e6i)9-s + 5.22e6·10-s − 1.60e7i·11-s + (−6.64e7 + 3.60e7i)12-s + 3.37e7·13-s − 1.07e8i·14-s + (2.41e7 + 4.45e7i)15-s + 3.59e8·16-s + 6.70e8i·17-s + ⋯
L(s)  = 1  − 1.76i·2-s + (0.879 − 0.476i)3-s − 2.10·4-s + 0.296i·5-s + (−0.840 − 1.55i)6-s + 0.580·7-s + 1.95i·8-s + (0.545 − 0.837i)9-s + 0.522·10-s − 0.825i·11-s + (−1.85 + 1.00i)12-s + 0.537·13-s − 1.02i·14-s + (0.141 + 0.260i)15-s + 1.33·16-s + 1.63i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.476i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.879 + 0.476i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3\)
Sign: $-0.879 + 0.476i$
Analytic conductor: \(3.72986\)
Root analytic conductor: \(1.93128\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{3} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3,\ (\ :7),\ -0.879 + 0.476i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.424187 - 1.67259i\)
\(L(\frac12)\) \(\approx\) \(0.424187 - 1.67259i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.92e3 + 1.04e3i)T \)
good2 \( 1 + 225. iT - 1.63e4T^{2} \)
5 \( 1 - 2.31e4iT - 6.10e9T^{2} \)
7 \( 1 - 4.78e5T + 6.78e11T^{2} \)
11 \( 1 + 1.60e7iT - 3.79e14T^{2} \)
13 \( 1 - 3.37e7T + 3.93e15T^{2} \)
17 \( 1 - 6.70e8iT - 1.68e17T^{2} \)
19 \( 1 - 3.92e8T + 7.99e17T^{2} \)
23 \( 1 - 3.24e7iT - 1.15e19T^{2} \)
29 \( 1 - 2.47e10iT - 2.97e20T^{2} \)
31 \( 1 + 3.17e10T + 7.56e20T^{2} \)
37 \( 1 + 4.38e10T + 9.01e21T^{2} \)
41 \( 1 + 1.11e11iT - 3.79e22T^{2} \)
43 \( 1 - 1.00e11T + 7.38e22T^{2} \)
47 \( 1 - 2.14e11iT - 2.56e23T^{2} \)
53 \( 1 + 5.81e11iT - 1.37e24T^{2} \)
59 \( 1 + 2.31e12iT - 6.19e24T^{2} \)
61 \( 1 + 4.22e12T + 9.87e24T^{2} \)
67 \( 1 + 2.32e12T + 3.67e25T^{2} \)
71 \( 1 - 1.42e13iT - 8.27e25T^{2} \)
73 \( 1 + 1.06e13T + 1.22e26T^{2} \)
79 \( 1 - 3.21e13T + 3.68e26T^{2} \)
83 \( 1 - 1.21e13iT - 7.36e26T^{2} \)
89 \( 1 + 3.00e13iT - 1.95e27T^{2} \)
97 \( 1 + 5.28e13T + 6.52e27T^{2} \)
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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

−21.82941932921279867464322380703, −20.61649012982825445842611794356, −19.30687849781800914280686213946, −18.17676535932356868420810643882, −14.29779699307371061375631772287, −12.76560516203029275033105041276, −10.84905188888106928659462293200, −8.690865224156601003655496090772, −3.43469876460585567545733956904, −1.49790629659222650011613731182, 4.79063690443691312602537021462, 7.56689236609126929954637885763, 9.192216863625731928744722271051, 13.72001322629477878391981229248, 15.05483191703219212729316129938, 16.34289105597545499246895964263, 18.17655795835580467566583922380, 20.61736783540586244420769435574, 22.71349657294131567188487235434, 24.43033538493855421497851368971

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