↑

Properties

Label 2-476-119.100-c1-0-0
Degree $2$
Conductor $476$
Sign $-0.998 + 0.0572i$
Analytic cond. $3.80087$
Root an. cond. $1.94958$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.940 + 1.22i)3-s + (0.338 − 2.57i)5-s + (−1.75 − 1.97i)7-s + (0.158 + 0.592i)9-s + (−4.68 + 0.616i)11-s + 3.87i·13-s + (2.83 + 2.83i)15-s + (−3.27 + 2.49i)17-s + (−6.62 + 1.77i)19-s + (4.07 − 0.298i)21-s + (−1.66 − 2.16i)23-s + (−1.68 − 0.451i)25-s + (−5.15 − 2.13i)27-s + (0.556 − 0.230i)29-s + (4.31 − 5.62i)31-s + ⋯
L(s)  = 1  + (−0.543 + 0.707i)3-s + (0.151 − 1.15i)5-s + (−0.665 − 0.746i)7-s + (0.0528 + 0.197i)9-s + (−1.41 + 0.186i)11-s + 1.07i·13-s + (0.732 + 0.732i)15-s + (−0.795 + 0.606i)17-s + (−1.51 + 0.407i)19-s + (0.889 − 0.0652i)21-s + (−0.346 − 0.451i)23-s + (−0.336 − 0.0902i)25-s + (−0.992 − 0.411i)27-s + (0.103 − 0.0428i)29-s + (0.775 − 1.01i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0572i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(476\)    =    \(2^{2} \cdot 7 \cdot 17\)
Sign: $-0.998 + 0.0572i$
Analytic conductor: \(3.80087\)
Root analytic conductor: \(1.94958\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{476} (457, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 476,\ (\ :1/2),\ -0.998 + 0.0572i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.000801771 - 0.0279652i\)
\(L(\frac12)\) \(\approx\) \(0.000801771 - 0.0279652i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (1.75 + 1.97i)T \)
17 \( 1 + (3.27 - 2.49i)T \)
good3 \( 1 + (0.940 - 1.22i)T + (-0.776 - 2.89i)T^{2} \)
5 \( 1 + (-0.338 + 2.57i)T + (-4.82 - 1.29i)T^{2} \)
11 \( 1 + (4.68 - 0.616i)T + (10.6 - 2.84i)T^{2} \)
13 \( 1 - 3.87iT - 13T^{2} \)
19 \( 1 + (6.62 - 1.77i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.66 + 2.16i)T + (-5.95 + 22.2i)T^{2} \)
29 \( 1 + (-0.556 + 0.230i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-4.31 + 5.62i)T + (-8.02 - 29.9i)T^{2} \)
37 \( 1 + (0.582 + 0.0767i)T + (35.7 + 9.57i)T^{2} \)
41 \( 1 + (2.37 + 0.983i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-2.76 + 2.76i)T - 43iT^{2} \)
47 \( 1 + (6.58 + 3.80i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.299 + 1.11i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-4.18 - 1.12i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.93 - 3.78i)T + (15.7 - 58.9i)T^{2} \)
67 \( 1 + (-4.69 - 8.13i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-2.13 - 5.14i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (11.5 + 8.83i)T + (18.8 + 70.5i)T^{2} \)
79 \( 1 + (-8.98 - 11.7i)T + (-20.4 + 76.3i)T^{2} \)
83 \( 1 + (7.85 + 7.85i)T + 83iT^{2} \)
89 \( 1 + (-1.39 - 0.804i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.18 + 2.14i)T + (68.5 - 68.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.25065108136688450472399747253, −10.41073718582092815729432871926, −9.941147491257308083901512721380, −8.816337740057683444836683768817, −8.014390225439293269341237548624, −6.71765677099911296742880503127, −5.69868277852780786716606226635, −4.56773476126131881722921759012, −4.17796526721909571668914145052, −2.13131841374912956444530893869, 0.01662003830731141878320007024, 2.40885191883332549884006049288, 3.18218620293421265391887346180, 5.06775240396509419227842600618, 6.16378583726273462949773444209, 6.61950207032494742151640092469, 7.62935175726852238244018786011, 8.661753743756063447931267935831, 9.893283119074538438194347482456, 10.64852323806582967811649986176

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