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Properties

Label 2-465-15.8-c1-0-8
Degree $2$
Conductor $465$
Sign $0.719 - 0.694i$
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  + (−1.54 − 1.54i)2-s + (−1.05 + 1.37i)3-s + 2.79i·4-s + (0.879 + 2.05i)5-s + (3.76 − 0.496i)6-s + (0.747 − 0.747i)7-s + (1.23 − 1.23i)8-s + (−0.778 − 2.89i)9-s + (1.82 − 4.54i)10-s − 4.27i·11-s + (−3.84 − 2.94i)12-s + (1.44 + 1.44i)13-s − 2.31·14-s + (−3.75 − 0.958i)15-s + 1.76·16-s + (1.93 + 1.93i)17-s + ⋯
L(s)  = 1  + (−1.09 − 1.09i)2-s + (−0.608 + 0.793i)3-s + 1.39i·4-s + (0.393 + 0.919i)5-s + (1.53 − 0.202i)6-s + (0.282 − 0.282i)7-s + (0.437 − 0.437i)8-s + (−0.259 − 0.965i)9-s + (0.576 − 1.43i)10-s − 1.28i·11-s + (−1.11 − 0.851i)12-s + (0.400 + 0.400i)13-s − 0.619·14-s + (−0.968 − 0.247i)15-s + 0.441·16-s + (0.469 + 0.469i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.719 - 0.694i$
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: \((2,\ 465,\ (\ :1/2),\ 0.719 - 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.555209 + 0.224080i\)
\(L(\frac12)\) \(\approx\) \(0.555209 + 0.224080i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.05 - 1.37i)T \)
5 \( 1 + (-0.879 - 2.05i)T \)
31 \( 1 + T \)
good2 \( 1 + (1.54 + 1.54i)T + 2iT^{2} \)
7 \( 1 + (-0.747 + 0.747i)T - 7iT^{2} \)
11 \( 1 + 4.27iT - 11T^{2} \)
13 \( 1 + (-1.44 - 1.44i)T + 13iT^{2} \)
17 \( 1 + (-1.93 - 1.93i)T + 17iT^{2} \)
19 \( 1 - 4.33iT - 19T^{2} \)
23 \( 1 + (5.15 - 5.15i)T - 23iT^{2} \)
29 \( 1 - 9.70T + 29T^{2} \)
37 \( 1 + (7.78 - 7.78i)T - 37iT^{2} \)
41 \( 1 - 0.692iT - 41T^{2} \)
43 \( 1 + (3.62 + 3.62i)T + 43iT^{2} \)
47 \( 1 + (-8.14 - 8.14i)T + 47iT^{2} \)
53 \( 1 + (-0.217 + 0.217i)T - 53iT^{2} \)
59 \( 1 + 4.28T + 59T^{2} \)
61 \( 1 - 2.56T + 61T^{2} \)
67 \( 1 + (1.87 - 1.87i)T - 67iT^{2} \)
71 \( 1 + 1.62iT - 71T^{2} \)
73 \( 1 + (-6.32 - 6.32i)T + 73iT^{2} \)
79 \( 1 - 3.86iT - 79T^{2} \)
83 \( 1 + (-5.30 + 5.30i)T - 83iT^{2} \)
89 \( 1 + 5.38T + 89T^{2} \)
97 \( 1 + (-5.06 + 5.06i)T - 97iT^{2} \)
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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

−10.85282638207367302083757568556, −10.36744930973315712235253532335, −9.773232751967160992986230825157, −8.739982486455971130383255851786, −7.912685548403832803774987034804, −6.38054751687706150688607459782, −5.61864514723940623618942976166, −3.83551097308456798785764548260, −3.08023595071142926384801841402, −1.37653207385464333167888296743, 0.63110078072435954793133701461, 2.05813571897410936239320172151, 4.75022448866718419612978150867, 5.54176697436971239021452857020, 6.51865303864199005315392037699, 7.27890150914664019541028598820, 8.220746733528962627576279151682, 8.792300697287237860765502343612, 9.864906648715777068346077917365, 10.58260886592351754337659956008

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