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Properties

Label 2-465-15.2-c1-0-2
Degree $2$
Conductor $465$
Sign $-0.786 + 0.618i$
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  + (0.195 − 0.195i)2-s + (−0.423 + 1.67i)3-s + 1.92i·4-s + (−2.09 + 0.787i)5-s + (0.245 + 0.411i)6-s + (−2.75 − 2.75i)7-s + (0.767 + 0.767i)8-s + (−2.64 − 1.42i)9-s + (−0.255 + 0.563i)10-s + 1.30i·11-s + (−3.23 − 0.815i)12-s + (3.21 − 3.21i)13-s − 1.07·14-s + (−0.435 − 3.84i)15-s − 3.54·16-s + (−1.44 + 1.44i)17-s + ⋯
L(s)  = 1  + (0.138 − 0.138i)2-s + (−0.244 + 0.969i)3-s + 0.961i·4-s + (−0.935 + 0.352i)5-s + (0.100 + 0.167i)6-s + (−1.04 − 1.04i)7-s + (0.271 + 0.271i)8-s + (−0.880 − 0.474i)9-s + (−0.0806 + 0.178i)10-s + 0.392i·11-s + (−0.932 − 0.235i)12-s + (0.892 − 0.892i)13-s − 0.288·14-s + (−0.112 − 0.993i)15-s − 0.886·16-s + (−0.349 + 0.349i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.786 + 0.618i$
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: \((2,\ 465,\ (\ :1/2),\ -0.786 + 0.618i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0740901 - 0.214117i\)
\(L(\frac12)\) \(\approx\) \(0.0740901 - 0.214117i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.423 - 1.67i)T \)
5 \( 1 + (2.09 - 0.787i)T \)
31 \( 1 + T \)
good2 \( 1 + (-0.195 + 0.195i)T - 2iT^{2} \)
7 \( 1 + (2.75 + 2.75i)T + 7iT^{2} \)
11 \( 1 - 1.30iT - 11T^{2} \)
13 \( 1 + (-3.21 + 3.21i)T - 13iT^{2} \)
17 \( 1 + (1.44 - 1.44i)T - 17iT^{2} \)
19 \( 1 - 1.91iT - 19T^{2} \)
23 \( 1 + (0.841 + 0.841i)T + 23iT^{2} \)
29 \( 1 + 10.6T + 29T^{2} \)
37 \( 1 + (2.30 + 2.30i)T + 37iT^{2} \)
41 \( 1 - 2.71iT - 41T^{2} \)
43 \( 1 + (-1.08 + 1.08i)T - 43iT^{2} \)
47 \( 1 + (6.09 - 6.09i)T - 47iT^{2} \)
53 \( 1 + (-2.73 - 2.73i)T + 53iT^{2} \)
59 \( 1 + 10.0T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 + (-5.05 - 5.05i)T + 67iT^{2} \)
71 \( 1 - 13.0iT - 71T^{2} \)
73 \( 1 + (5.56 - 5.56i)T - 73iT^{2} \)
79 \( 1 - 7.10iT - 79T^{2} \)
83 \( 1 + (2.09 + 2.09i)T + 83iT^{2} \)
89 \( 1 - 13.6T + 89T^{2} \)
97 \( 1 + (0.122 + 0.122i)T + 97iT^{2} \)
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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

−11.31413110713746470851623323210, −10.83997867830143594955660427199, −9.987928742769726890343607875821, −8.902191670570025975826189756524, −7.920632995812139230394168540945, −7.11814102156265373862046006800, −5.99500162676458657435127753378, −4.39274513059018501801132969919, −3.72528772308721151297925166970, −3.14136237480672477154096863669, 0.13264556306761338770076814052, 1.84526480917531041491756992884, 3.42710514124087966338916048490, 4.96836071347469599580463454310, 5.97996167102159070367093535954, 6.56977018543161724855151476239, 7.57742284826005767800229344843, 8.922099141686607268379227169695, 9.202864851876081549218444168157, 10.81731929490008800835302079719

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