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Properties

Label 2-464-29.25-c1-0-2
Degree $2$
Conductor $464$
Sign $0.181 - 0.983i$
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  + (−0.5 + 0.240i)3-s + (−0.0440 + 0.193i)5-s + (0.0990 − 0.0476i)7-s + (−1.67 + 2.10i)9-s + (0.832 + 1.04i)11-s + (2.45 + 3.07i)13-s + (−0.0244 − 0.107i)15-s − 2.91·17-s + (1.16 + 0.562i)19-s + (−0.0380 + 0.0476i)21-s + (1.73 + 7.59i)23-s + (4.46 + 2.15i)25-s + (0.702 − 3.07i)27-s + (−1.39 − 5.20i)29-s + (−2.07 + 9.11i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.139i)3-s + (−0.0197 + 0.0863i)5-s + (0.0374 − 0.0180i)7-s + (−0.559 + 0.701i)9-s + (0.250 + 0.314i)11-s + (0.681 + 0.854i)13-s + (−0.00631 − 0.0276i)15-s − 0.706·17-s + (0.267 + 0.128i)19-s + (−0.00829 + 0.0104i)21-s + (0.361 + 1.58i)23-s + (0.893 + 0.430i)25-s + (0.135 − 0.592i)27-s + (−0.259 − 0.965i)29-s + (−0.373 + 1.63i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(464\)    =    \(2^{4} \cdot 29\)
Sign: $0.181 - 0.983i$
Analytic conductor: \(3.70505\)
Root analytic conductor: \(1.92485\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{464} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 464,\ (\ :1/2),\ 0.181 - 0.983i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859243 + 0.714887i\)
\(L(\frac12)\) \(\approx\) \(0.859243 + 0.714887i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.39 + 5.20i)T \)
good3 \( 1 + (0.5 - 0.240i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 + (0.0440 - 0.193i)T + (-4.50 - 2.16i)T^{2} \)
7 \( 1 + (-0.0990 + 0.0476i)T + (4.36 - 5.47i)T^{2} \)
11 \( 1 + (-0.832 - 1.04i)T + (-2.44 + 10.7i)T^{2} \)
13 \( 1 + (-2.45 - 3.07i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 + 2.91T + 17T^{2} \)
19 \( 1 + (-1.16 - 0.562i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.73 - 7.59i)T + (-20.7 + 9.97i)T^{2} \)
31 \( 1 + (2.07 - 9.11i)T + (-27.9 - 13.4i)T^{2} \)
37 \( 1 + (1.88 - 2.36i)T + (-8.23 - 36.0i)T^{2} \)
41 \( 1 - 3.76T + 41T^{2} \)
43 \( 1 + (1.48 + 6.49i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (0.5 + 0.626i)T + (-10.4 + 45.8i)T^{2} \)
53 \( 1 + (1.85 - 8.12i)T + (-47.7 - 22.9i)T^{2} \)
59 \( 1 - 5.08T + 59T^{2} \)
61 \( 1 + (-9.96 + 4.79i)T + (38.0 - 47.6i)T^{2} \)
67 \( 1 + (-6.85 + 8.60i)T + (-14.9 - 65.3i)T^{2} \)
71 \( 1 + (6.82 + 8.56i)T + (-15.7 + 69.2i)T^{2} \)
73 \( 1 + (1.76 + 7.74i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (3.04 - 3.82i)T + (-17.5 - 77.0i)T^{2} \)
83 \( 1 + (10.5 + 5.06i)T + (51.7 + 64.8i)T^{2} \)
89 \( 1 + (-2.55 + 11.2i)T + (-80.1 - 38.6i)T^{2} \)
97 \( 1 + (-9.54 - 4.59i)T + (60.4 + 75.8i)T^{2} \)
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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

−11.24977893111382058940231170840, −10.53605721622573148261230680874, −9.365549853914604637721276266472, −8.669140320544647393693338799474, −7.52574939911643988516479810841, −6.60838362258801639760301570710, −5.52745946152659668962514775040, −4.57912405901694332082131399160, −3.31383315789870859867158623464, −1.75380060967562116397895823470, 0.74477705444360168192915131000, 2.72238204261923733085909339015, 3.92381720454737345785994018263, 5.24483757121904171981532850110, 6.19626560628650350382532614233, 6.96954000646607479067387707204, 8.375454258351649974455048847187, 8.844582835048466936974927013869, 10.01317776505009755562911142573, 11.07026152714439790698454192276

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