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Properties

Label 2-464-116.115-c2-0-22
Degree $2$
Conductor $464$
Sign $-0.939 - 0.343i$
Analytic cond. $12.6430$
Root an. cond. $3.55571$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5.21·3-s − 2.51·5-s − 9.89i·7-s + 18.1·9-s + 1.53·11-s + 13.6·13-s + 13.1·15-s − 9.31i·17-s − 7.44·19-s + 51.5i·21-s − 25.8i·23-s − 18.6·25-s − 47.8·27-s + (4.98 + 28.5i)29-s − 29.4·31-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.503·5-s − 1.41i·7-s + 2.01·9-s + 0.139·11-s + 1.05·13-s + 0.875·15-s − 0.547i·17-s − 0.391·19-s + 2.45i·21-s − 1.12i·23-s − 0.746·25-s − 1.77·27-s + (0.171 + 0.985i)29-s − 0.950·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.939 - 0.343i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $-0.939 - 0.343i$
Analytic conductor: \(12.6430\)
Root analytic conductor: \(3.55571\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (463, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1),\ -0.939 - 0.343i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.1568427321\)
\(L(\frac12)\) \(\approx\) \(0.1568427321\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-4.98 - 28.5i)T \)
good3 \( 1 + 5.21T + 9T^{2} \)
5 \( 1 + 2.51T + 25T^{2} \)
7 \( 1 + 9.89iT - 49T^{2} \)
11 \( 1 - 1.53T + 121T^{2} \)
13 \( 1 - 13.6T + 169T^{2} \)
17 \( 1 + 9.31iT - 289T^{2} \)
19 \( 1 + 7.44T + 361T^{2} \)
23 \( 1 + 25.8iT - 529T^{2} \)
31 \( 1 + 29.4T + 961T^{2} \)
37 \( 1 - 2.13iT - 1.36e3T^{2} \)
41 \( 1 + 20.2iT - 1.68e3T^{2} \)
43 \( 1 + 71.6T + 1.84e3T^{2} \)
47 \( 1 - 61.9T + 2.20e3T^{2} \)
53 \( 1 + 70.5T + 2.80e3T^{2} \)
59 \( 1 - 31.0iT - 3.48e3T^{2} \)
61 \( 1 + 96.2iT - 3.72e3T^{2} \)
67 \( 1 - 122. iT - 4.48e3T^{2} \)
71 \( 1 - 116. iT - 5.04e3T^{2} \)
73 \( 1 - 17.2iT - 5.32e3T^{2} \)
79 \( 1 + 75.8T + 6.24e3T^{2} \)
83 \( 1 - 9.58iT - 6.88e3T^{2} \)
89 \( 1 - 125. iT - 7.92e3T^{2} \)
97 \( 1 + 3.83iT - 9.40e3T^{2} \)
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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

−10.75571750451650515486422462163, −9.832150968203473076208753871633, −8.391786097625283195100102241281, −7.17283215555586043342751670233, −6.67941934810548515481447391397, −5.60280310946624739592439058536, −4.51347114383860194498804963383, −3.76466548483116894860472685401, −1.21100939756268596298436002768, −0.094416267035342889613228516824, 1.68310813553043020611671278926, 3.67835349232632605757884400911, 4.88152462853033355216248907073, 5.91825671873340498683718730350, 6.20619079929748063037987913291, 7.56962418819708044534177808027, 8.651767936355870790566793336045, 9.684424227214758333133842713012, 10.76236915012883044530505322339, 11.47895231926138213493481036479

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