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Properties

Label 2-465-465.464-c1-0-12
Degree $2$
Conductor $465$
Sign $-0.683 - 0.729i$
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  + 1.41·2-s + (−1.5 + 0.866i)3-s + (1.41 + 1.73i)5-s + (−2.12 + 1.22i)6-s + 2.44i·7-s − 2.82·8-s + (1.5 − 2.59i)9-s + (2.00 + 2.44i)10-s − 4.24·11-s + 3.46i·14-s + (−3.62 − 1.37i)15-s − 4.00·16-s + 1.73i·17-s + (2.12 − 3.67i)18-s − 19-s + ⋯
L(s)  = 1  + 1.00·2-s + (−0.866 + 0.499i)3-s + (0.632 + 0.774i)5-s + (−0.866 + 0.499i)6-s + 0.925i·7-s − 0.999·8-s + (0.5 − 0.866i)9-s + (0.632 + 0.774i)10-s − 1.27·11-s + 0.925i·14-s + (−0.935 − 0.354i)15-s − 1.00·16-s + 0.420i·17-s + (0.499 − 0.866i)18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 - 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $-0.683 - 0.729i$
Analytic conductor: \(3.71304\)
Root analytic conductor: \(1.92692\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{465} (464, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 465,\ (\ :1/2),\ -0.683 - 0.729i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488687 + 1.12743i\)
\(L(\frac12)\) \(\approx\) \(0.488687 + 1.12743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (-1.41 - 1.73i)T \)
31 \( 1 + (5 + 2.44i)T \)
good2 \( 1 - 1.41T + 2T^{2} \)
7 \( 1 - 2.44iT - 7T^{2} \)
11 \( 1 + 4.24T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 1.73iT - 17T^{2} \)
19 \( 1 + T + 19T^{2} \)
23 \( 1 - 6.92iT - 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 + 1.73iT - 41T^{2} \)
43 \( 1 - 9T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 - 8.66iT - 53T^{2} \)
59 \( 1 - 1.73iT - 59T^{2} \)
61 \( 1 + 9.79iT - 61T^{2} \)
67 \( 1 + 7.34iT - 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + 3T + 73T^{2} \)
79 \( 1 + 2.44iT - 79T^{2} \)
83 \( 1 + 1.73iT - 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 4.89iT - 97T^{2} \)
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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

−11.40359890214793686578243849882, −10.65466412653943714869029777924, −9.727432081334997665540514796738, −8.956367551019387164822821163538, −7.44405634927717377158534191282, −6.07494286464426842858975542391, −5.72371630712474767768126830081, −4.90143356290354438870512202919, −3.61597811530844152662773706597, −2.48817030099019428700650332475, 0.59608295567762393972429543406, 2.49967174654146032635644146557, 4.29382971525236152049031128251, 4.96264023085114821488600139634, 5.74289761465911865867311158741, 6.67029574059283600554393070929, 7.79774070189195816038988813034, 8.876406654021781894161980085962, 10.13944129107790266155472606841, 10.73663168976210347247846736549

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