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Properties

Label 2-465-15.2-c1-0-12
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: $\chi_{465} (32, \cdot )$
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

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

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