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Properties

Label 2-465-15.8-c1-0-35
Degree $2$
Conductor $465$
Sign $0.689 + 0.724i$
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  + (0.523 + 0.523i)2-s + (−0.695 + 1.58i)3-s − 1.45i·4-s + (−2.20 + 0.349i)5-s + (−1.19 + 0.466i)6-s + (2.00 − 2.00i)7-s + (1.80 − 1.80i)8-s + (−2.03 − 2.20i)9-s + (−1.34 − 0.973i)10-s − 2.51i·11-s + (2.30 + 1.00i)12-s + (−1.36 − 1.36i)13-s + 2.09·14-s + (0.981 − 3.74i)15-s − 1.00·16-s + (−2.74 − 2.74i)17-s + ⋯
L(s)  = 1  + (0.370 + 0.370i)2-s + (−0.401 + 0.915i)3-s − 0.725i·4-s + (−0.987 + 0.156i)5-s + (−0.488 + 0.190i)6-s + (0.757 − 0.757i)7-s + (0.639 − 0.639i)8-s + (−0.677 − 0.735i)9-s + (−0.423 − 0.307i)10-s − 0.756i·11-s + (0.664 + 0.291i)12-s + (−0.378 − 0.378i)13-s + 0.560·14-s + (0.253 − 0.967i)15-s − 0.251·16-s + (−0.666 − 0.666i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.689 + 0.724i$
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: \((2,\ 465,\ (\ :1/2),\ 0.689 + 0.724i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04732 - 0.448899i\)
\(L(\frac12)\) \(\approx\) \(1.04732 - 0.448899i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.695 - 1.58i)T \)
5 \( 1 + (2.20 - 0.349i)T \)
31 \( 1 + T \)
good2 \( 1 + (-0.523 - 0.523i)T + 2iT^{2} \)
7 \( 1 + (-2.00 + 2.00i)T - 7iT^{2} \)
11 \( 1 + 2.51iT - 11T^{2} \)
13 \( 1 + (1.36 + 1.36i)T + 13iT^{2} \)
17 \( 1 + (2.74 + 2.74i)T + 17iT^{2} \)
19 \( 1 - 4.33iT - 19T^{2} \)
23 \( 1 + (-6.46 + 6.46i)T - 23iT^{2} \)
29 \( 1 - 5.35T + 29T^{2} \)
37 \( 1 + (1.27 - 1.27i)T - 37iT^{2} \)
41 \( 1 - 0.543iT - 41T^{2} \)
43 \( 1 + (4.32 + 4.32i)T + 43iT^{2} \)
47 \( 1 + (-1.52 - 1.52i)T + 47iT^{2} \)
53 \( 1 + (6.26 - 6.26i)T - 53iT^{2} \)
59 \( 1 + 3.16T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (-7.77 + 7.77i)T - 67iT^{2} \)
71 \( 1 + 7.13iT - 71T^{2} \)
73 \( 1 + (6.43 + 6.43i)T + 73iT^{2} \)
79 \( 1 - 7.24iT - 79T^{2} \)
83 \( 1 + (5.53 - 5.53i)T - 83iT^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (-10.2 + 10.2i)T - 97iT^{2} \)
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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

−10.68943199710082341138737749868, −10.51429097874389545004192087353, −9.165232869173023276322991875098, −8.173636436430315749241741831203, −7.08327840840557960015830520012, −6.15111179147533093526835566304, −4.84716606545955067059406621610, −4.53068788291348199938905008388, −3.24903398988228511069189559417, −0.68041925794521437924429395995, 1.81312883410395144527202511397, 2.99639280130992465640652730773, 4.52712772149292623359104961325, 5.13653109111043458788498826480, 6.81721024104153248324878781546, 7.46222106429616084285533012690, 8.316266803925396290440574897185, 9.009523031910384566712580452242, 10.84836553822104656467445533468, 11.56938122840324918076377637111

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