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Properties

Label 2-465-15.2-c1-0-22
Degree $2$
Conductor $465$
Sign $0.444 + 0.895i$
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  + (−1.93 + 1.93i)2-s + (−1.43 − 0.963i)3-s − 5.47i·4-s + (−1.96 + 1.06i)5-s + (4.64 − 0.921i)6-s + (1.12 + 1.12i)7-s + (6.72 + 6.72i)8-s + (1.14 + 2.77i)9-s + (1.73 − 5.86i)10-s − 0.704i·11-s + (−5.27 + 7.88i)12-s + (−4.82 + 4.82i)13-s − 4.35·14-s + (3.85 + 0.357i)15-s − 15.0·16-s + (1.83 − 1.83i)17-s + ⋯
L(s)  = 1  + (−1.36 + 1.36i)2-s + (−0.831 − 0.556i)3-s − 2.73i·4-s + (−0.878 + 0.476i)5-s + (1.89 − 0.376i)6-s + (0.425 + 0.425i)7-s + (2.37 + 2.37i)8-s + (0.381 + 0.924i)9-s + (0.549 − 1.85i)10-s − 0.212i·11-s + (−1.52 + 2.27i)12-s + (−1.33 + 1.33i)13-s − 1.16·14-s + (0.995 + 0.0923i)15-s − 3.76·16-s + (0.445 − 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.444 + 0.895i$
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: \((2,\ 465,\ (\ :1/2),\ 0.444 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0934983 - 0.0579513i\)
\(L(\frac12)\) \(\approx\) \(0.0934983 - 0.0579513i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.43 + 0.963i)T \)
5 \( 1 + (1.96 - 1.06i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.93 - 1.93i)T - 2iT^{2} \)
7 \( 1 + (-1.12 - 1.12i)T + 7iT^{2} \)
11 \( 1 + 0.704iT - 11T^{2} \)
13 \( 1 + (4.82 - 4.82i)T - 13iT^{2} \)
17 \( 1 + (-1.83 + 1.83i)T - 17iT^{2} \)
19 \( 1 + 1.74iT - 19T^{2} \)
23 \( 1 + (-2.24 - 2.24i)T + 23iT^{2} \)
29 \( 1 + 4.83T + 29T^{2} \)
37 \( 1 + (6.24 + 6.24i)T + 37iT^{2} \)
41 \( 1 + 5.49iT - 41T^{2} \)
43 \( 1 + (-0.999 + 0.999i)T - 43iT^{2} \)
47 \( 1 + (5.28 - 5.28i)T - 47iT^{2} \)
53 \( 1 + (2.76 + 2.76i)T + 53iT^{2} \)
59 \( 1 - 1.42T + 59T^{2} \)
61 \( 1 - 4.88T + 61T^{2} \)
67 \( 1 + (6.87 + 6.87i)T + 67iT^{2} \)
71 \( 1 + 14.6iT - 71T^{2} \)
73 \( 1 + (-5.99 + 5.99i)T - 73iT^{2} \)
79 \( 1 - 11.7iT - 79T^{2} \)
83 \( 1 + (8.42 + 8.42i)T + 83iT^{2} \)
89 \( 1 + 12.0T + 89T^{2} \)
97 \( 1 + (-0.0249 - 0.0249i)T + 97iT^{2} \)
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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

−10.85841681205766005397957959502, −9.761704650988012434417166985335, −8.902503462670895357428617847837, −7.83447931747163138682921338651, −7.22429122041064435401009087213, −6.71569465753714653988399046079, −5.51248488040104434113950084738, −4.72237244663378442961844408332, −1.95082496754204634173743718326, −0.13821098195820716746411965732, 1.16210976426727712056122929865, 3.08716227753969171760194263684, 4.12676527403785079306497225151, 5.10023342212695068477562664159, 7.13124768389324967000933536381, 7.86579035572912570254180747517, 8.662203408388309552723737326786, 9.860464128539731526521917896452, 10.22952601929114196075713251267, 11.12412406108608166662831945647

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