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Properties

Label 2-465-15.8-c1-0-39
Degree $2$
Conductor $465$
Sign $0.999 + 0.0389i$
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  + (1.36 + 1.36i)2-s + (0.324 − 1.70i)3-s + 1.70i·4-s + (−1.70 + 1.44i)5-s + (2.75 − 1.87i)6-s + (2.86 − 2.86i)7-s + (0.405 − 0.405i)8-s + (−2.78 − 1.10i)9-s + (−4.28 − 0.353i)10-s − 5.71i·11-s + (2.89 + 0.552i)12-s + (3.46 + 3.46i)13-s + 7.78·14-s + (1.90 + 3.37i)15-s + 4.50·16-s + (0.490 + 0.490i)17-s + ⋯
L(s)  = 1  + (0.962 + 0.962i)2-s + (0.187 − 0.982i)3-s + 0.850i·4-s + (−0.762 + 0.646i)5-s + (1.12 − 0.764i)6-s + (1.08 − 1.08i)7-s + (0.143 − 0.143i)8-s + (−0.929 − 0.368i)9-s + (−1.35 − 0.111i)10-s − 1.72i·11-s + (0.835 + 0.159i)12-s + (0.959 + 0.959i)13-s + 2.08·14-s + (0.492 + 0.870i)15-s + 1.12·16-s + (0.119 + 0.119i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.999 + 0.0389i$
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: \((2,\ 465,\ (\ :1/2),\ 0.999 + 0.0389i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41327 - 0.0470162i\)
\(L(\frac12)\) \(\approx\) \(2.41327 - 0.0470162i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.324 + 1.70i)T \)
5 \( 1 + (1.70 - 1.44i)T \)
31 \( 1 - T \)
good2 \( 1 + (-1.36 - 1.36i)T + 2iT^{2} \)
7 \( 1 + (-2.86 + 2.86i)T - 7iT^{2} \)
11 \( 1 + 5.71iT - 11T^{2} \)
13 \( 1 + (-3.46 - 3.46i)T + 13iT^{2} \)
17 \( 1 + (-0.490 - 0.490i)T + 17iT^{2} \)
19 \( 1 - 5.62iT - 19T^{2} \)
23 \( 1 + (3.70 - 3.70i)T - 23iT^{2} \)
29 \( 1 + 4.36T + 29T^{2} \)
37 \( 1 + (-2.13 + 2.13i)T - 37iT^{2} \)
41 \( 1 + 3.32iT - 41T^{2} \)
43 \( 1 + (0.639 + 0.639i)T + 43iT^{2} \)
47 \( 1 + (-1.35 - 1.35i)T + 47iT^{2} \)
53 \( 1 + (2.53 - 2.53i)T - 53iT^{2} \)
59 \( 1 - 0.0212T + 59T^{2} \)
61 \( 1 + 2.98T + 61T^{2} \)
67 \( 1 + (10.3 - 10.3i)T - 67iT^{2} \)
71 \( 1 - 11.8iT - 71T^{2} \)
73 \( 1 + (7.20 + 7.20i)T + 73iT^{2} \)
79 \( 1 - 6.29iT - 79T^{2} \)
83 \( 1 + (-4.54 + 4.54i)T - 83iT^{2} \)
89 \( 1 + 5.57T + 89T^{2} \)
97 \( 1 + (-5.70 + 5.70i)T - 97iT^{2} \)
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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

−11.29788378637512209711102423541, −10.52052217215931529262538567438, −8.613509063804632806749051622224, −7.84534684696558664162932899339, −7.38714291246756859650381542352, −6.31616447430202865378050346924, −5.72124720215739233248216313346, −4.06747521254235140588143190154, −3.54114336158938212959148582807, −1.32635491084163946448312336722, 1.97846126613816560729261816191, 3.15085962612845321800592157489, 4.46379507819949615080932403637, 4.73973191594286837589656620084, 5.67191392979811326115894055863, 7.74736401163599973358402143815, 8.435083955117642405299520072858, 9.362502899659655601733424324498, 10.49013210970731950961976753587, 11.27219529211336979403685631134

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