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Properties

Label 2-369-9.4-c1-0-30
Degree $2$
Conductor $369$
Sign $0.654 + 0.755i$
Analytic cond. $2.94647$
Root an. cond. $1.71653$
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  + (−0.111 − 0.193i)2-s + (1.67 + 0.435i)3-s + (0.974 − 1.68i)4-s + (0.0946 − 0.163i)5-s + (−0.103 − 0.373i)6-s + (−1.37 − 2.38i)7-s − 0.883·8-s + (2.62 + 1.45i)9-s − 0.0423·10-s + (0.463 + 0.803i)11-s + (2.36 − 2.40i)12-s + (0.893 − 1.54i)13-s + (−0.308 + 0.533i)14-s + (0.230 − 0.233i)15-s + (−1.85 − 3.20i)16-s + 2.65·17-s + ⋯
L(s)  = 1  + (−0.0790 − 0.136i)2-s + (0.967 + 0.251i)3-s + (0.487 − 0.844i)4-s + (0.0423 − 0.0733i)5-s + (−0.0421 − 0.152i)6-s + (−0.520 − 0.901i)7-s − 0.312·8-s + (0.873 + 0.486i)9-s − 0.0133·10-s + (0.139 + 0.242i)11-s + (0.683 − 0.694i)12-s + (0.247 − 0.429i)13-s + (−0.0823 + 0.142i)14-s + (0.0594 − 0.0603i)15-s + (−0.462 − 0.801i)16-s + 0.644·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.654 + 0.755i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.654 + 0.755i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.70828 - 0.780201i\)
\(L(\frac12)\) \(\approx\) \(1.70828 - 0.780201i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.67 - 0.435i)T \)
41 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.111 + 0.193i)T + (-1 + 1.73i)T^{2} \)
5 \( 1 + (-0.0946 + 0.163i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (1.37 + 2.38i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.463 - 0.803i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-0.893 + 1.54i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 2.65T + 17T^{2} \)
19 \( 1 + 4.39T + 19T^{2} \)
23 \( 1 + (0.924 - 1.60i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.42 - 4.20i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.348 - 0.604i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 1.90T + 37T^{2} \)
43 \( 1 + (0.658 + 1.14i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.839 - 1.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.10T + 53T^{2} \)
59 \( 1 + (5.56 - 9.64i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.0536 - 0.0929i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.64 - 2.85i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 1.77T + 71T^{2} \)
73 \( 1 - 0.140T + 73T^{2} \)
79 \( 1 + (0.487 + 0.845i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.45 - 9.45i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 17.7T + 89T^{2} \)
97 \( 1 + (-5.81 - 10.0i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.83347504623366850118770621990, −10.35877194919686663385734632882, −9.582781348589565778639429355629, −8.663608918343568543776860864484, −7.43458803801651078417442356980, −6.69790225192492934053899890373, −5.36909888763919898719623178402, −4.04164787421682389720783340754, −2.90064245173521981866323302833, −1.39492095488772618712099243434, 2.19568236206555445299749512523, 3.09063027186109103218369928071, 4.21081509071750222710070168001, 6.13337919235245244845717671795, 6.80431113429715636857883543780, 8.001851975149609734498393002119, 8.595908022100378640472152050223, 9.391740152638058934406988037356, 10.53902408398552205075844449539, 11.84419022326951812743506569662

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