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Properties

Label 2-465-15.2-c1-0-26
Degree $2$
Conductor $465$
Sign $0.461 - 0.887i$
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  + (0.450 − 0.450i)2-s + (−0.451 + 1.67i)3-s + 1.59i·4-s + (1.55 − 1.60i)5-s + (0.549 + 0.955i)6-s + (3.64 + 3.64i)7-s + (1.61 + 1.61i)8-s + (−2.59 − 1.50i)9-s + (−0.0230 − 1.42i)10-s − 5.56i·11-s + (−2.66 − 0.719i)12-s + (0.744 − 0.744i)13-s + 3.27·14-s + (1.98 + 3.32i)15-s − 1.73·16-s + (−2.78 + 2.78i)17-s + ⋯
L(s)  = 1  + (0.318 − 0.318i)2-s + (−0.260 + 0.965i)3-s + 0.797i·4-s + (0.695 − 0.718i)5-s + (0.224 + 0.390i)6-s + (1.37 + 1.37i)7-s + (0.572 + 0.572i)8-s + (−0.864 − 0.503i)9-s + (−0.00728 − 0.450i)10-s − 1.67i·11-s + (−0.769 − 0.207i)12-s + (0.206 − 0.206i)13-s + 0.876·14-s + (0.512 + 0.858i)15-s − 0.433·16-s + (−0.674 + 0.674i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 465 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.461 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(465\)    =    \(3 \cdot 5 \cdot 31\)
Sign: $0.461 - 0.887i$
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: \((2,\ 465,\ (\ :1/2),\ 0.461 - 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61242 + 0.979290i\)
\(L(\frac12)\) \(\approx\) \(1.61242 + 0.979290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.451 - 1.67i)T \)
5 \( 1 + (-1.55 + 1.60i)T \)
31 \( 1 + T \)
good2 \( 1 + (-0.450 + 0.450i)T - 2iT^{2} \)
7 \( 1 + (-3.64 - 3.64i)T + 7iT^{2} \)
11 \( 1 + 5.56iT - 11T^{2} \)
13 \( 1 + (-0.744 + 0.744i)T - 13iT^{2} \)
17 \( 1 + (2.78 - 2.78i)T - 17iT^{2} \)
19 \( 1 - 4.67iT - 19T^{2} \)
23 \( 1 + (-1.93 - 1.93i)T + 23iT^{2} \)
29 \( 1 + 0.0153T + 29T^{2} \)
37 \( 1 + (5.59 + 5.59i)T + 37iT^{2} \)
41 \( 1 - 0.377iT - 41T^{2} \)
43 \( 1 + (-5.72 + 5.72i)T - 43iT^{2} \)
47 \( 1 + (-1.44 + 1.44i)T - 47iT^{2} \)
53 \( 1 + (2.79 + 2.79i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 + (2.34 + 2.34i)T + 67iT^{2} \)
71 \( 1 - 0.837iT - 71T^{2} \)
73 \( 1 + (-1.19 + 1.19i)T - 73iT^{2} \)
79 \( 1 + 7.83iT - 79T^{2} \)
83 \( 1 + (-0.130 - 0.130i)T + 83iT^{2} \)
89 \( 1 - 0.746T + 89T^{2} \)
97 \( 1 + (-1.44 - 1.44i)T + 97iT^{2} \)
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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

−11.20388400159858101452428787885, −10.65202692151439069578789543410, −9.074263855753422329421525285948, −8.662114339446074310842041957026, −8.082429996092803970674268369172, −5.89898407477842979431156443356, −5.48169170592712103670155215506, −4.47634336742338965986212873819, −3.32401558682667184764004573384, −1.98230347726081177276136617821, 1.28334077875421265907359476036, 2.24694314119637027861759985350, 4.54493090098870259739676564478, 5.08382586148230292134289328896, 6.52177120722987264383976881962, 7.04893693391556284095093019409, 7.60277682366380499370821920309, 9.167742810159117144456935014805, 10.26998164730604154590886427942, 10.90085973653846879535716273706

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