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Properties

Label 2-464-116.99-c1-0-7
Degree $2$
Conductor $464$
Sign $0.957 - 0.288i$
Analytic cond. $3.70505$
Root an. cond. $1.92485$
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  + (−1.92 + 1.92i)3-s − 2.39i·5-s + 1.01i·7-s − 4.44i·9-s + (2.07 − 2.07i)11-s − 1.30i·13-s + (4.62 + 4.62i)15-s + (3.78 + 3.78i)17-s + (3.70 − 3.70i)19-s + (−1.96 − 1.96i)21-s + 4.28i·23-s − 0.746·25-s + (2.79 + 2.79i)27-s + (3.15 + 4.36i)29-s + (−2.93 + 2.93i)31-s + ⋯
L(s)  = 1  + (−1.11 + 1.11i)3-s − 1.07i·5-s + 0.385i·7-s − 1.48i·9-s + (0.624 − 0.624i)11-s − 0.361i·13-s + (1.19 + 1.19i)15-s + (0.917 + 0.917i)17-s + (0.849 − 0.849i)19-s + (−0.429 − 0.429i)21-s + 0.894i·23-s − 0.149·25-s + (0.537 + 0.537i)27-s + (0.585 + 0.810i)29-s + (−0.527 + 0.527i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.957 - 0.288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.957 - 0.288i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (447, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ 0.957 - 0.288i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00294 + 0.147669i\)
\(L(\frac12)\) \(\approx\) \(1.00294 + 0.147669i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
29 \( 1 + (-3.15 - 4.36i)T \)
good3 \( 1 + (1.92 - 1.92i)T - 3iT^{2} \)
5 \( 1 + 2.39iT - 5T^{2} \)
7 \( 1 - 1.01iT - 7T^{2} \)
11 \( 1 + (-2.07 + 2.07i)T - 11iT^{2} \)
13 \( 1 + 1.30iT - 13T^{2} \)
17 \( 1 + (-3.78 - 3.78i)T + 17iT^{2} \)
19 \( 1 + (-3.70 + 3.70i)T - 19iT^{2} \)
23 \( 1 - 4.28iT - 23T^{2} \)
31 \( 1 + (2.93 - 2.93i)T - 31iT^{2} \)
37 \( 1 + (1.63 - 1.63i)T - 37iT^{2} \)
41 \( 1 + (-8.28 + 8.28i)T - 41iT^{2} \)
43 \( 1 + (-8.05 + 8.05i)T - 43iT^{2} \)
47 \( 1 + (-4.09 - 4.09i)T + 47iT^{2} \)
53 \( 1 - 6.32T + 53T^{2} \)
59 \( 1 - 4.26iT - 59T^{2} \)
61 \( 1 + (8.36 + 8.36i)T + 61iT^{2} \)
67 \( 1 + 9.62T + 67T^{2} \)
71 \( 1 + 11.5T + 71T^{2} \)
73 \( 1 + (2.47 - 2.47i)T - 73iT^{2} \)
79 \( 1 + (-10.5 + 10.5i)T - 79iT^{2} \)
83 \( 1 + 0.320iT - 83T^{2} \)
89 \( 1 + (8.21 + 8.21i)T + 89iT^{2} \)
97 \( 1 + (-1.53 + 1.53i)T - 97iT^{2} \)
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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

−10.96335145919676565082273191815, −10.36498463559711939578804462212, −9.172191824879672424223601679517, −8.862067507479546081325681586536, −7.39369832683007947532979196646, −5.82071229954035039410882835800, −5.50300186812009185383965587811, −4.48908971524345596172284796639, −3.43971916179367268002275191192, −0.989503998671296186057152787187, 1.11271067324653218719776058036, 2.67073765808992756340683290509, 4.26505426645619006848664728158, 5.67883287700071369773553417086, 6.43391740138368443795629132869, 7.26085747500021313465546173449, 7.69703098989160339823779500692, 9.424881205777763256386376626409, 10.34397442814046781479393126324, 11.15939441127963934510814077884

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