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Properties

Label 2-465-93.92-c1-0-40
Degree $2$
Conductor $465$
Sign $-0.969 + 0.246i$
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  − 0.201i·2-s + (−0.798 − 1.53i)3-s + 1.95·4-s − i·5-s + (−0.310 + 0.161i)6-s − 3.41·7-s − 0.798i·8-s + (−1.72 + 2.45i)9-s − 0.201·10-s − 5.52·11-s + (−1.56 − 3.01i)12-s − 5.36i·13-s + 0.688i·14-s + (−1.53 + 0.798i)15-s + 3.75·16-s − 3.84·17-s + ⋯
L(s)  = 1  − 0.142i·2-s + (−0.461 − 0.887i)3-s + 0.979·4-s − 0.447i·5-s + (−0.126 + 0.0658i)6-s − 1.29·7-s − 0.282i·8-s + (−0.574 + 0.818i)9-s − 0.0638·10-s − 1.66·11-s + (−0.451 − 0.869i)12-s − 1.48i·13-s + 0.184i·14-s + (−0.396 + 0.206i)15-s + 0.939·16-s − 0.932·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.246i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.969 + 0.246i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.969 + 0.246i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0992048 - 0.791097i\)
\(L(\frac12)\) \(\approx\) \(0.0992048 - 0.791097i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.798 + 1.53i)T \)
5 \( 1 + iT \)
31 \( 1 + (5.42 + 1.26i)T \)
good2 \( 1 + 0.201iT - 2T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 5.52T + 11T^{2} \)
13 \( 1 + 5.36iT - 13T^{2} \)
17 \( 1 + 3.84T + 17T^{2} \)
19 \( 1 - 3.81T + 19T^{2} \)
23 \( 1 - 0.602T + 23T^{2} \)
29 \( 1 - 3.16T + 29T^{2} \)
37 \( 1 - 0.169iT - 37T^{2} \)
41 \( 1 + 8.16iT - 41T^{2} \)
43 \( 1 - 2.74iT - 43T^{2} \)
47 \( 1 + 12.0iT - 47T^{2} \)
53 \( 1 - 9.95T + 53T^{2} \)
59 \( 1 + 3.14iT - 59T^{2} \)
61 \( 1 - 8.27iT - 61T^{2} \)
67 \( 1 + 9.06T + 67T^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 - 5.65iT - 73T^{2} \)
79 \( 1 + 16.0iT - 79T^{2} \)
83 \( 1 + 4.72T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−10.54439248582857233232846460167, −10.17879684057934567858737875811, −8.638209070895262931578325151508, −7.60881931621540169454358514618, −7.03487459237599102831127132610, −5.87192537808161046632631657620, −5.33451973085971916317400394355, −3.19928155270375030059607157252, −2.35017851442048297421178109297, −0.46382806770174470402813002956, 2.53034391607838767455966624407, 3.42397979300873722188092913842, 4.84981521938021710813627435266, 6.01808951633818813711866083531, 6.65061653648344391099660715670, 7.56334972898605192620211288907, 9.017162978415159835551221477505, 9.850533655466321078593302223785, 10.61371276799686682457914072192, 11.23904373051796961892182858514

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