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Properties

Label 2-465-15.8-c1-0-53
Degree $2$
Conductor $465$
Sign $-0.988 - 0.150i$
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  + (−0.963 − 0.963i)2-s + (−0.212 − 1.71i)3-s − 0.144i·4-s + (2.14 − 0.615i)5-s + (−1.45 + 1.86i)6-s + (0.630 − 0.630i)7-s + (−2.06 + 2.06i)8-s + (−2.90 + 0.731i)9-s + (−2.66 − 1.47i)10-s − 5.14i·11-s + (−0.248 + 0.0307i)12-s + (2.18 + 2.18i)13-s − 1.21·14-s + (−1.51 − 3.56i)15-s + 3.69·16-s + (−5.34 − 5.34i)17-s + ⋯
L(s)  = 1  + (−0.681 − 0.681i)2-s + (−0.122 − 0.992i)3-s − 0.0722i·4-s + (0.961 − 0.275i)5-s + (−0.592 + 0.759i)6-s + (0.238 − 0.238i)7-s + (−0.730 + 0.730i)8-s + (−0.969 + 0.243i)9-s + (−0.842 − 0.467i)10-s − 1.55i·11-s + (−0.0716 + 0.00887i)12-s + (0.606 + 0.606i)13-s − 0.324·14-s + (−0.391 − 0.920i)15-s + 0.922·16-s + (−1.29 − 1.29i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.988 - 0.150i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.988 - 0.150i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.988 - 0.150i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0705696 + 0.929362i\)
\(L(\frac12)\) \(\approx\) \(0.0705696 + 0.929362i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.212 + 1.71i)T \)
5 \( 1 + (-2.14 + 0.615i)T \)
31 \( 1 - T \)
good2 \( 1 + (0.963 + 0.963i)T + 2iT^{2} \)
7 \( 1 + (-0.630 + 0.630i)T - 7iT^{2} \)
11 \( 1 + 5.14iT - 11T^{2} \)
13 \( 1 + (-2.18 - 2.18i)T + 13iT^{2} \)
17 \( 1 + (5.34 + 5.34i)T + 17iT^{2} \)
19 \( 1 + 0.608iT - 19T^{2} \)
23 \( 1 + (2.85 - 2.85i)T - 23iT^{2} \)
29 \( 1 - 4.66T + 29T^{2} \)
37 \( 1 + (2.43 - 2.43i)T - 37iT^{2} \)
41 \( 1 - 0.351iT - 41T^{2} \)
43 \( 1 + (-0.242 - 0.242i)T + 43iT^{2} \)
47 \( 1 + (6.60 + 6.60i)T + 47iT^{2} \)
53 \( 1 + (9.16 - 9.16i)T - 53iT^{2} \)
59 \( 1 - 14.0T + 59T^{2} \)
61 \( 1 + 1.43T + 61T^{2} \)
67 \( 1 + (-7.99 + 7.99i)T - 67iT^{2} \)
71 \( 1 + 8.14iT - 71T^{2} \)
73 \( 1 + (-7.92 - 7.92i)T + 73iT^{2} \)
79 \( 1 + 2.49iT - 79T^{2} \)
83 \( 1 + (-3.84 + 3.84i)T - 83iT^{2} \)
89 \( 1 - 4.88T + 89T^{2} \)
97 \( 1 + (2.39 - 2.39i)T - 97iT^{2} \)
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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

−10.81099622555024402500357290805, −9.577866008205477952648453211931, −8.830200932848366624320653112581, −8.224097225294920868583249490092, −6.68864929233836400270635749041, −6.03963090454267016139285002182, −5.04194093826146225475739411516, −2.94313006950305199101546963816, −1.85306905730521446472877285158, −0.72714684751839076134513617199, 2.26748758621218316484141555600, 3.77671476515853430428957321026, 4.93678324872966533515786914406, 6.14373258347223637549552162478, 6.74905938666943334301471437849, 8.195325913842644040520408509191, 8.754625753724540591207552216331, 9.790205199449329429502805813236, 10.20185654362512605631519659646, 11.18184438520109017290460131922

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