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Properties

Label 2-351-9.4-c1-0-9
Degree $2$
Conductor $351$
Sign $-0.561 + 0.827i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
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  + (−0.248 − 0.429i)2-s + (0.876 − 1.51i)4-s + (−0.663 + 1.14i)5-s + (−1.56 − 2.71i)7-s − 1.86·8-s + 0.658·10-s + (−1.70 − 2.95i)11-s + (0.5 − 0.866i)13-s + (−0.776 + 1.34i)14-s + (−1.29 − 2.23i)16-s + 0.912·17-s − 1.03·19-s + (1.16 + 2.01i)20-s + (−0.847 + 1.46i)22-s + (2.68 − 4.65i)23-s + ⋯
L(s)  = 1  + (−0.175 − 0.303i)2-s + (0.438 − 0.759i)4-s + (−0.296 + 0.513i)5-s + (−0.591 − 1.02i)7-s − 0.658·8-s + 0.208·10-s + (−0.514 − 0.891i)11-s + (0.138 − 0.240i)13-s + (−0.207 + 0.359i)14-s + (−0.322 − 0.559i)16-s + 0.221·17-s − 0.237·19-s + (0.260 + 0.450i)20-s + (−0.180 + 0.312i)22-s + (0.560 − 0.970i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.561 + 0.827i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.561 + 0.827i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $-0.561 + 0.827i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (118, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ -0.561 + 0.827i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.456620 - 0.862260i\)
\(L(\frac12)\) \(\approx\) \(0.456620 - 0.862260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.248 + 0.429i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (0.663 - 1.14i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.56 + 2.71i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.70 + 2.95i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 0.912T + 17T^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
23 \( 1 + (-2.68 + 4.65i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.25 + 5.63i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.27 - 2.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 7.99T + 37T^{2} \)
41 \( 1 + (2.72 - 4.72i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.20 + 3.81i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4.59 + 7.96i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 + (3.54 - 6.13i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.18 - 12.4i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4.67 + 8.10i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 5.22T + 73T^{2} \)
79 \( 1 + (-6.05 - 10.4i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.50 - 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 - 8.33T + 89T^{2} \)
97 \( 1 + (7.01 + 12.1i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.91526299183717647345112369134, −10.46444022458989571887770548325, −9.620288373764409000467139877397, −8.377066631742823820108274545113, −7.19028170834114586000108578840, −6.44771514112647651130634647514, −5.36773591181553492907497863564, −3.76157639641518824757915946138, −2.67122670436137824212277810882, −0.69354828103442112418227886128, 2.29229402106037376648181184589, 3.51495533998659693431081336284, 4.94840084056686396242954554679, 6.12288761630756389879401839348, 7.15673393307078583155827975073, 8.043105370506459131397629268244, 8.944008548032117308632124088404, 9.666519461627505816760584095777, 11.07613302451081148214314192289, 11.94889674503438593362590245696

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