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Properties

Label 2-465-15.8-c1-0-33
Degree $2$
Conductor $465$
Sign $0.989 + 0.146i$
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  + (1.04 + 1.04i)2-s + (−1.52 + 0.828i)3-s + 0.164i·4-s + (0.204 − 2.22i)5-s + (−2.44 − 0.720i)6-s + (−1.82 + 1.82i)7-s + (1.90 − 1.90i)8-s + (1.62 − 2.51i)9-s + (2.52 − 2.10i)10-s − 5.69i·11-s + (−0.136 − 0.250i)12-s + (4.14 + 4.14i)13-s − 3.80·14-s + (1.53 + 3.55i)15-s + 4.30·16-s + (−0.349 − 0.349i)17-s + ⋯
L(s)  = 1  + (0.735 + 0.735i)2-s + (−0.878 + 0.478i)3-s + 0.0824i·4-s + (0.0915 − 0.995i)5-s + (−0.997 − 0.294i)6-s + (−0.690 + 0.690i)7-s + (0.675 − 0.675i)8-s + (0.542 − 0.839i)9-s + (0.799 − 0.665i)10-s − 1.71i·11-s + (−0.0394 − 0.0724i)12-s + (1.14 + 1.14i)13-s − 1.01·14-s + (0.395 + 0.918i)15-s + 1.07·16-s + (−0.0848 − 0.0848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.989 + 0.146i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.989 + 0.146i$
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: \((2,\ 465,\ (\ :1/2),\ 0.989 + 0.146i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53120 - 0.112502i\)
\(L(\frac12)\) \(\approx\) \(1.53120 - 0.112502i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.52 - 0.828i)T \)
5 \( 1 + (-0.204 + 2.22i)T \)
31 \( 1 - T \)
good2 \( 1 + (-1.04 - 1.04i)T + 2iT^{2} \)
7 \( 1 + (1.82 - 1.82i)T - 7iT^{2} \)
11 \( 1 + 5.69iT - 11T^{2} \)
13 \( 1 + (-4.14 - 4.14i)T + 13iT^{2} \)
17 \( 1 + (0.349 + 0.349i)T + 17iT^{2} \)
19 \( 1 + 4.02iT - 19T^{2} \)
23 \( 1 + (-3.77 + 3.77i)T - 23iT^{2} \)
29 \( 1 - 4.44T + 29T^{2} \)
37 \( 1 + (2.31 - 2.31i)T - 37iT^{2} \)
41 \( 1 - 7.89iT - 41T^{2} \)
43 \( 1 + (3.86 + 3.86i)T + 43iT^{2} \)
47 \( 1 + (7.41 + 7.41i)T + 47iT^{2} \)
53 \( 1 + (-2.50 + 2.50i)T - 53iT^{2} \)
59 \( 1 + 6.80T + 59T^{2} \)
61 \( 1 - 11.1T + 61T^{2} \)
67 \( 1 + (5.48 - 5.48i)T - 67iT^{2} \)
71 \( 1 - 1.37iT - 71T^{2} \)
73 \( 1 + (-4.49 - 4.49i)T + 73iT^{2} \)
79 \( 1 - 0.679iT - 79T^{2} \)
83 \( 1 + (-6.41 + 6.41i)T - 83iT^{2} \)
89 \( 1 + 8.86T + 89T^{2} \)
97 \( 1 + (-7.46 + 7.46i)T - 97iT^{2} \)
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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

−11.22153385244195452704607392313, −10.14195739907137702495655051442, −9.065322944577650022011430423200, −8.550702456642997308124142651601, −6.60338353389652192115625074857, −6.28733947598311160422455060454, −5.36908230916638417294444398428, −4.61146282115914720550927553247, −3.49576588904632072069859564380, −0.927623691428764138122155577169, 1.68918555324991233221654288234, 3.11182993545953173665567661094, 4.07239180050328083955291257793, 5.27970438886410786343200417864, 6.38557717080120824612662338577, 7.23925082474659493384207895566, 7.945767494664779292350896611806, 9.903505337959337957584919665894, 10.48304262872107665216062918306, 11.06437933395008082355210287678

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