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Properties

Label 2-465-465.254-c1-0-24
Degree $2$
Conductor $465$
Sign $0.999 + 0.000719i$
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  − 2.61·2-s + (1.5 − 0.866i)3-s + 4.85·4-s + (−1.11 + 1.93i)5-s + (−3.92 + 2.26i)6-s − 7.47·8-s + (1.5 − 2.59i)9-s + (2.92 − 5.06i)10-s + (7.28 − 4.20i)12-s + 3.87i·15-s + 9.85·16-s + (6.35 − 3.66i)17-s + (−3.92 + 6.80i)18-s + (2.35 + 4.07i)19-s + (−5.42 + 9.39i)20-s + ⋯
L(s)  = 1  − 1.85·2-s + (0.866 − 0.499i)3-s + 2.42·4-s + (−0.499 + 0.866i)5-s + (−1.60 + 0.925i)6-s − 2.64·8-s + (0.5 − 0.866i)9-s + (0.925 − 1.60i)10-s + (2.10 − 1.21i)12-s + 1.00i·15-s + 2.46·16-s + (1.54 − 0.889i)17-s + (−0.925 + 1.60i)18-s + (0.540 + 0.935i)19-s + (−1.21 + 2.10i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.000719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.000719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.999 + 0.000719i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (254, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ 0.999 + 0.000719i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.798634 - 0.000287356i\)
\(L(\frac12)\) \(\approx\) \(0.798634 - 0.000287356i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.5 + 0.866i)T \)
5 \( 1 + (1.11 - 1.93i)T \)
31 \( 1 + (-4 - 3.87i)T \)
good2 \( 1 + 2.61T + 2T^{2} \)
7 \( 1 + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-6.35 + 3.66i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.35 - 4.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 3.46iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
37 \( 1 + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 13.4T + 47T^{2} \)
53 \( 1 + (-12 - 6.92i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 9.47iT - 61T^{2} \)
67 \( 1 + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-15.3 + 8.86i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (14.9 + 8.61i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.65797789020241444392840990022, −9.942742851178892991598564314178, −9.259789600296464196057772107712, −8.206324639720022731956705156131, −7.57435206588545513916780533890, −7.11886202439262669788636555184, −5.97098382003047487445037243950, −3.48844340518373593595468814286, −2.61400673863914626657821254179, −1.15632190428874774905344454738, 1.06392684881715801462541517504, 2.55139144419489316253408119160, 3.91965885965044618307453507341, 5.45918973602223957355988541782, 7.02966620444303347061194907151, 7.908710387023186247723065264520, 8.413330293251860221209668506332, 9.158555742004003434526957100807, 9.902083457557329310151306890043, 10.59049506873456378842535521515

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