Properties

Label 2-465-465.119-c1-0-45
Degree $2$
Conductor $465$
Sign $0.856 - 0.516i$
Analytic cond. $3.71304$
Root an. cond. $1.92692$
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  + 2.44·2-s + (−0.178 + 1.72i)3-s + 3.99·4-s + (1.93 − 1.11i)5-s + (−0.436 + 4.22i)6-s + 4.89·8-s + (−2.93 − 0.614i)9-s + (4.74 − 2.73i)10-s + (−1.93 − 3.35i)11-s + (−0.712 + 6.89i)12-s + (1.58 + 2.73i)13-s + (1.58 + 3.53i)15-s + 3.99·16-s + (−4.89 − 2.82i)17-s + (−7.19 − 1.50i)18-s + (−1 + 1.73i)19-s + ⋯
L(s)  = 1  + 1.73·2-s + (−0.102 + 0.994i)3-s + 1.99·4-s + (0.866 − 0.499i)5-s + (−0.178 + 1.72i)6-s + 1.73·8-s + (−0.978 − 0.204i)9-s + (1.50 − 0.866i)10-s + (−0.583 − 1.01i)11-s + (−0.205 + 1.98i)12-s + (0.438 + 0.759i)13-s + (0.408 + 0.912i)15-s + 0.999·16-s + (−1.18 − 0.685i)17-s + (−1.69 − 0.354i)18-s + (−0.229 + 0.397i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.856 - 0.516i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (119, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.856 - 0.516i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.59172 + 0.999858i\)
\(L(\frac12)\) \(\approx\) \(3.59172 + 0.999858i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.178 - 1.72i)T \)
5 \( 1 + (-1.93 + 1.11i)T \)
31 \( 1 + (3.5 - 4.33i)T \)
good2 \( 1 - 2.44T + 2T^{2} \)
7 \( 1 + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.93 + 3.35i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.58 - 2.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (4.89 + 2.82i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
37 \( 1 + (3.16 - 5.47i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-9.68 + 5.59i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.58 + 2.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 9.79T + 47T^{2} \)
53 \( 1 + (-4.89 + 2.82i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.93 - 1.11i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 - 1.73iT - 61T^{2} \)
67 \( 1 + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.93 - 1.11i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.58 - 2.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.5 - 2.59i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.57 + 4.94i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + 11.6T + 89T^{2} \)
97 \( 1 + 10.9iT - 97T^{2} \)
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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

−11.20857595860480283395509692819, −10.62834747774695022333678925159, −9.359181532123229361396818033633, −8.638769123833120583827504941990, −6.92943628402959079102168462305, −5.80953780199219155268210038501, −5.41777668230731040859990168310, −4.40840464898628282996046310417, −3.48947982135789709242328711602, −2.26654821321878134095280655221, 2.08608280139705332378166087237, 2.69863526886279417386427284531, 4.24154700669189483488181694634, 5.43950747727214203973392211470, 6.12480990584231870222660171465, 6.85848190451284993879420028614, 7.73019351958804979891962774596, 9.154224600202450017849546602548, 10.78107062314127937298182814315, 10.95912117597424701617137202648

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