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Properties

Label 2-369-41.2-c1-0-8
Degree $2$
Conductor $369$
Sign $0.853 - 0.520i$
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  + (1.40 − 0.457i)2-s + (0.153 − 0.111i)4-s + (0.455 + 0.626i)5-s + (2.12 + 4.16i)7-s + (−1.57 + 2.16i)8-s + (0.926 + 0.673i)10-s + (0.538 − 0.0852i)11-s + (−0.808 − 0.411i)13-s + (4.89 + 4.89i)14-s + (−1.34 + 4.13i)16-s + (−0.937 − 5.91i)17-s + (3.90 − 1.98i)19-s + (0.139 + 0.0452i)20-s + (0.718 − 0.366i)22-s + (0.323 + 0.995i)23-s + ⋯
L(s)  = 1  + (0.995 − 0.323i)2-s + (0.0765 − 0.0556i)4-s + (0.203 + 0.280i)5-s + (0.802 + 1.57i)7-s + (−0.556 + 0.766i)8-s + (0.293 + 0.212i)10-s + (0.162 − 0.0257i)11-s + (−0.224 − 0.114i)13-s + (1.30 + 1.30i)14-s + (−0.335 + 1.03i)16-s + (−0.227 − 1.43i)17-s + (0.895 − 0.456i)19-s + (0.0311 + 0.0101i)20-s + (0.153 − 0.0780i)22-s + (0.0674 + 0.207i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369\)    =    \(3^{2} \cdot 41\)
Sign: $0.853 - 0.520i$
Analytic conductor: \(2.94647\)
Root analytic conductor: \(1.71653\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{369} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 369,\ (\ :1/2),\ 0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10211 + 0.590290i\)
\(L(\frac12)\) \(\approx\) \(2.10211 + 0.590290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
41 \( 1 + (6.21 - 1.55i)T \)
good2 \( 1 + (-1.40 + 0.457i)T + (1.61 - 1.17i)T^{2} \)
5 \( 1 + (-0.455 - 0.626i)T + (-1.54 + 4.75i)T^{2} \)
7 \( 1 + (-2.12 - 4.16i)T + (-4.11 + 5.66i)T^{2} \)
11 \( 1 + (-0.538 + 0.0852i)T + (10.4 - 3.39i)T^{2} \)
13 \( 1 + (0.808 + 0.411i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.937 + 5.91i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (-3.90 + 1.98i)T + (11.1 - 15.3i)T^{2} \)
23 \( 1 + (-0.323 - 0.995i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.193 - 1.21i)T + (-27.5 - 8.96i)T^{2} \)
31 \( 1 + (-1.22 - 0.893i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-5.87 + 4.26i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (5.91 - 1.92i)T + (34.7 - 25.2i)T^{2} \)
47 \( 1 + (-1.80 + 3.53i)T + (-27.6 - 38.0i)T^{2} \)
53 \( 1 + (0.837 - 5.28i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (2.74 + 8.44i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.31 - 0.428i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (5.56 + 0.881i)T + (63.7 + 20.7i)T^{2} \)
71 \( 1 + (-1.07 + 0.170i)T + (67.5 - 21.9i)T^{2} \)
73 \( 1 + 5.61iT - 73T^{2} \)
79 \( 1 + (-8.62 + 8.62i)T - 79iT^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + (-0.885 - 1.73i)T + (-52.3 + 72.0i)T^{2} \)
97 \( 1 + (1.91 + 0.303i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.84699238343777335476914522092, −11.00455676975986636432096097547, −9.496799697171440641549874995257, −8.825168131377403298917725285252, −7.79239695338728613200281022784, −6.36310195846770785578564658582, −5.26409855080745568791330000742, −4.79085173660706660357141390530, −3.11278376041511804622173951561, −2.28705763275525643364423605355, 1.30304456838026118680096087435, 3.58127825886942134215393391840, 4.38667419560619029916104379226, 5.25510224277468263757139480219, 6.40472565751760734841035852662, 7.35491346531459182594907234739, 8.358034486055348985105039446108, 9.655945280649156417756588447173, 10.42432483332998671671841878255, 11.42870904250849650482860049414

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