Properties

Label 2-465-465.254-c1-0-7
Degree $2$
Conductor $465$
Sign $-0.978 - 0.208i$
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  + 1.15·2-s + (−0.150 + 1.72i)3-s − 0.658·4-s + (−1.19 − 1.89i)5-s + (−0.174 + 1.99i)6-s + (0.189 − 0.109i)7-s − 3.07·8-s + (−2.95 − 0.519i)9-s + (−1.38 − 2.18i)10-s + (−3.23 + 5.60i)11-s + (0.0990 − 1.13i)12-s + (−1.94 + 3.37i)13-s + (0.220 − 0.127i)14-s + (3.44 − 1.77i)15-s − 2.24·16-s + (−1.53 + 0.889i)17-s + ⋯
L(s)  = 1  + 0.819·2-s + (−0.0868 + 0.996i)3-s − 0.329·4-s + (−0.534 − 0.845i)5-s + (−0.0711 + 0.815i)6-s + (0.0718 − 0.0414i)7-s − 1.08·8-s + (−0.984 − 0.173i)9-s + (−0.437 − 0.692i)10-s + (−0.975 + 1.68i)11-s + (0.0285 − 0.327i)12-s + (−0.540 + 0.935i)13-s + (0.0588 − 0.0339i)14-s + (0.888 − 0.458i)15-s − 0.562·16-s + (−0.373 + 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.978 - 0.208i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.978 - 0.208i$
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: \((2,\ 465,\ (\ :1/2),\ -0.978 - 0.208i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0635554 + 0.602680i\)
\(L(\frac12)\) \(\approx\) \(0.0635554 + 0.602680i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.150 - 1.72i)T \)
5 \( 1 + (1.19 + 1.89i)T \)
31 \( 1 + (-5.53 + 0.642i)T \)
good2 \( 1 - 1.15T + 2T^{2} \)
7 \( 1 + (-0.189 + 0.109i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (3.23 - 5.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.94 - 3.37i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.53 - 0.889i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.113 - 0.195i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 6.66iT - 23T^{2} \)
29 \( 1 - 3.61T + 29T^{2} \)
37 \( 1 + (-1.21 - 2.10i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.613 - 0.354i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (4.15 + 7.18i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.68T + 47T^{2} \)
53 \( 1 + (-7.94 - 4.58i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (7.20 - 4.15i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 10.0iT - 61T^{2} \)
67 \( 1 + (-0.362 - 0.209i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4.13 - 2.38i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.75 + 8.24i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (7.00 - 4.04i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (-2.68 - 1.55i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.11T + 89T^{2} \)
97 \( 1 + 3.68iT - 97T^{2} \)
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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

−11.81646465279293022975852562588, −10.44810231597908734137075773026, −9.688770445056521401253720466406, −8.881461357799583460769578090979, −8.025004544588483596323901894248, −6.61618752707543172731206949852, −5.22544190043962596325753849330, −4.56505693647052346280613583394, −4.23028174361833818068493154966, −2.62036225986113931697096366410, 0.27784436753951717675103045225, 2.82091586919885768209732977250, 3.34018720992843124700741355032, 5.05115143497986568980808095390, 5.84155932841941985908338089163, 6.73515455787536931132542380331, 8.004116461114962931253245810060, 8.321776746666800221856417913676, 9.837233377845552356803255667390, 11.06502647454755568209362507117

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