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Properties

Label 2-464-29.16-c1-0-11
Degree $2$
Conductor $464$
Sign $0.734 + 0.678i$
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  + (1.52 − 1.90i)3-s + (2.60 + 1.25i)5-s + (1.89 − 2.37i)7-s + (−0.658 − 2.88i)9-s + (−1.22 + 5.34i)11-s + (0.0239 − 0.104i)13-s + (6.34 − 3.05i)15-s + 0.816·17-s + (−1.27 − 1.59i)19-s + (−1.65 − 7.24i)21-s + (−8.25 + 3.97i)23-s + (2.07 + 2.60i)25-s + (0.0938 + 0.0451i)27-s + (−5.37 + 0.304i)29-s + (−3.10 − 1.49i)31-s + ⋯
L(s)  = 1  + (0.878 − 1.10i)3-s + (1.16 + 0.559i)5-s + (0.717 − 0.899i)7-s + (−0.219 − 0.961i)9-s + (−0.367 + 1.61i)11-s + (0.00663 − 0.0290i)13-s + (1.63 − 0.789i)15-s + 0.197·17-s + (−0.292 − 0.366i)19-s + (−0.360 − 1.58i)21-s + (−1.72 + 0.829i)23-s + (0.414 + 0.520i)25-s + (0.0180 + 0.00869i)27-s + (−0.998 + 0.0564i)29-s + (−0.556 − 0.268i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 + 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.734 + 0.678i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ 0.734 + 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.10004 - 0.821254i\)
\(L(\frac12)\) \(\approx\) \(2.10004 - 0.821254i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.37 - 0.304i)T \)
good3 \( 1 + (-1.52 + 1.90i)T + (-0.667 - 2.92i)T^{2} \)
5 \( 1 + (-2.60 - 1.25i)T + (3.11 + 3.90i)T^{2} \)
7 \( 1 + (-1.89 + 2.37i)T + (-1.55 - 6.82i)T^{2} \)
11 \( 1 + (1.22 - 5.34i)T + (-9.91 - 4.77i)T^{2} \)
13 \( 1 + (-0.0239 + 0.104i)T + (-11.7 - 5.64i)T^{2} \)
17 \( 1 - 0.816T + 17T^{2} \)
19 \( 1 + (1.27 + 1.59i)T + (-4.22 + 18.5i)T^{2} \)
23 \( 1 + (8.25 - 3.97i)T + (14.3 - 17.9i)T^{2} \)
31 \( 1 + (3.10 + 1.49i)T + (19.3 + 24.2i)T^{2} \)
37 \( 1 + (1.31 + 5.75i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + 2.43T + 41T^{2} \)
43 \( 1 + (-4.94 + 2.37i)T + (26.8 - 33.6i)T^{2} \)
47 \( 1 + (-1.31 + 5.76i)T + (-42.3 - 20.3i)T^{2} \)
53 \( 1 + (3.55 + 1.71i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + 1.13T + 59T^{2} \)
61 \( 1 + (-5.09 + 6.38i)T + (-13.5 - 59.4i)T^{2} \)
67 \( 1 + (-0.212 - 0.932i)T + (-60.3 + 29.0i)T^{2} \)
71 \( 1 + (-0.531 + 2.32i)T + (-63.9 - 30.8i)T^{2} \)
73 \( 1 + (8.01 - 3.85i)T + (45.5 - 57.0i)T^{2} \)
79 \( 1 + (0.934 + 4.09i)T + (-71.1 + 34.2i)T^{2} \)
83 \( 1 + (-9.64 - 12.0i)T + (-18.4 + 80.9i)T^{2} \)
89 \( 1 + (4.99 + 2.40i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-7.43 - 9.32i)T + (-21.5 + 94.5i)T^{2} \)
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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

−10.74220476946993983476673634276, −10.01285064936278243818754352571, −9.195569337251926700182398700538, −7.81851340327382223815877566715, −7.45200403601865274386816313552, −6.58376558866066493920567852649, −5.34363250586619089385337175724, −3.92234992552661092441227100797, −2.22689421326857065778402454036, −1.79828297139175314417048987032, 1.94295983287534860523239579831, 3.09355067933264740337005005412, 4.36433969746639456174217829797, 5.51068268022931471805444974768, 6.01276299926821425384571061737, 8.089331209950532711734438225610, 8.596159847691078721660484109427, 9.271839805039208377629551010624, 10.08688065191009583796382045234, 10.88781100498986220634827033822

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