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Properties

Label 2-465-15.2-c1-0-25
Degree $2$
Conductor $465$
Sign $0.460 - 0.887i$
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  + (−1.04 + 1.04i)2-s + (1.71 − 0.211i)3-s − 0.203i·4-s + (−0.904 − 2.04i)5-s + (−1.58 + 2.02i)6-s + (2.89 + 2.89i)7-s + (−1.88 − 1.88i)8-s + (2.91 − 0.727i)9-s + (3.09 + 1.19i)10-s + 1.70i·11-s + (−0.0429 − 0.349i)12-s + (0.569 − 0.569i)13-s − 6.07·14-s + (−1.98 − 3.32i)15-s + 4.36·16-s + (0.889 − 0.889i)17-s + ⋯
L(s)  = 1  + (−0.742 + 0.742i)2-s + (0.992 − 0.122i)3-s − 0.101i·4-s + (−0.404 − 0.914i)5-s + (−0.645 + 0.827i)6-s + (1.09 + 1.09i)7-s + (−0.666 − 0.666i)8-s + (0.970 − 0.242i)9-s + (0.978 + 0.378i)10-s + 0.513i·11-s + (−0.0124 − 0.100i)12-s + (0.157 − 0.157i)13-s − 1.62·14-s + (−0.513 − 0.858i)15-s + 1.09·16-s + (0.215 − 0.215i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.460 - 0.887i$
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: \((2,\ 465,\ (\ :1/2),\ 0.460 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19805 + 0.728345i\)
\(L(\frac12)\) \(\approx\) \(1.19805 + 0.728345i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.71 + 0.211i)T \)
5 \( 1 + (0.904 + 2.04i)T \)
31 \( 1 - T \)
good2 \( 1 + (1.04 - 1.04i)T - 2iT^{2} \)
7 \( 1 + (-2.89 - 2.89i)T + 7iT^{2} \)
11 \( 1 - 1.70iT - 11T^{2} \)
13 \( 1 + (-0.569 + 0.569i)T - 13iT^{2} \)
17 \( 1 + (-0.889 + 0.889i)T - 17iT^{2} \)
19 \( 1 + 0.591iT - 19T^{2} \)
23 \( 1 + (-5.81 - 5.81i)T + 23iT^{2} \)
29 \( 1 - 2.29T + 29T^{2} \)
37 \( 1 + (6.19 + 6.19i)T + 37iT^{2} \)
41 \( 1 - 10.1iT - 41T^{2} \)
43 \( 1 + (-4.70 + 4.70i)T - 43iT^{2} \)
47 \( 1 + (1.16 - 1.16i)T - 47iT^{2} \)
53 \( 1 + (3.71 + 3.71i)T + 53iT^{2} \)
59 \( 1 + 4.33T + 59T^{2} \)
61 \( 1 + 10.5T + 61T^{2} \)
67 \( 1 + (-1.47 - 1.47i)T + 67iT^{2} \)
71 \( 1 - 1.54iT - 71T^{2} \)
73 \( 1 + (-5.44 + 5.44i)T - 73iT^{2} \)
79 \( 1 - 9.07iT - 79T^{2} \)
83 \( 1 + (5.94 + 5.94i)T + 83iT^{2} \)
89 \( 1 + 5.71T + 89T^{2} \)
97 \( 1 + (13.2 + 13.2i)T + 97iT^{2} \)
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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

−11.27243644189267171412436821982, −9.668168215037414986366303526322, −9.064754201621706323998993801605, −8.451974221088982882102696222911, −7.82500771310936976320042167824, −7.08768268563418978539030665991, −5.55969328913100983933102633024, −4.51238329401377002455610729518, −3.11455760482493771274212419703, −1.51923262993765551394174556139, 1.23530368107551091997876897787, 2.58982380920271950102698843330, 3.63261804878311147679428683396, 4.79844976107624658739627146853, 6.55419456581109948512625097610, 7.59634854004902415834699048697, 8.296725227628965795763971325729, 9.079084777397904555018160470855, 10.35235110002404399049293901896, 10.59899202598864519586334175739

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