↑

Properties

Label 2-475-475.111-c1-0-23
Degree $2$
Conductor $475$
Sign $0.723 - 0.690i$
Analytic cond. $3.79289$
Root an. cond. $1.94753$
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  + (1.95 + 0.136i)2-s + (−0.0107 − 0.308i)3-s + (1.83 + 0.258i)4-s + (−0.891 + 2.05i)5-s + (0.0211 − 0.606i)6-s + (1.49 + 2.59i)7-s + (−0.275 − 0.0585i)8-s + (2.89 − 0.202i)9-s + (−2.02 + 3.89i)10-s + (−0.0475 − 0.452i)11-s + (0.0599 − 0.570i)12-s + (0.885 + 1.31i)13-s + (2.57 + 5.28i)14-s + (0.642 + 0.253i)15-s + (−4.10 − 1.17i)16-s + (0.654 − 1.61i)17-s + ⋯
L(s)  = 1  + (1.38 + 0.0968i)2-s + (−0.00622 − 0.178i)3-s + (0.919 + 0.129i)4-s + (−0.398 + 0.917i)5-s + (0.00864 − 0.247i)6-s + (0.565 + 0.979i)7-s + (−0.0974 − 0.0207i)8-s + (0.965 − 0.0675i)9-s + (−0.641 + 1.23i)10-s + (−0.0143 − 0.136i)11-s + (0.0173 − 0.164i)12-s + (0.245 + 0.364i)13-s + (0.688 + 1.41i)14-s + (0.165 + 0.0653i)15-s + (−1.02 − 0.293i)16-s + (0.158 − 0.392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.723 - 0.690i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.723 - 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.60648 + 1.04395i\)
\(L(\frac12)\) \(\approx\) \(2.60648 + 1.04395i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.891 - 2.05i)T \)
19 \( 1 + (-3.63 - 2.40i)T \)
good2 \( 1 + (-1.95 - 0.136i)T + (1.98 + 0.278i)T^{2} \)
3 \( 1 + (0.0107 + 0.308i)T + (-2.99 + 0.209i)T^{2} \)
7 \( 1 + (-1.49 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.0475 + 0.452i)T + (-10.7 + 2.28i)T^{2} \)
13 \( 1 + (-0.885 - 1.31i)T + (-4.86 + 12.0i)T^{2} \)
17 \( 1 + (-0.654 + 1.61i)T + (-12.2 - 11.8i)T^{2} \)
23 \( 1 + (2.34 + 2.26i)T + (0.802 + 22.9i)T^{2} \)
29 \( 1 + (-0.323 - 0.801i)T + (-20.8 + 20.1i)T^{2} \)
31 \( 1 + (-2.41 + 2.68i)T + (-3.24 - 30.8i)T^{2} \)
37 \( 1 + (1.72 + 1.25i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-1.75 - 0.503i)T + (34.7 + 21.7i)T^{2} \)
43 \( 1 + (-0.372 + 2.11i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (3.93 + 9.74i)T + (-33.8 + 32.6i)T^{2} \)
53 \( 1 + (1.63 + 0.229i)T + (50.9 + 14.6i)T^{2} \)
59 \( 1 + (0.240 + 0.965i)T + (-52.0 + 27.6i)T^{2} \)
61 \( 1 + (3.78 + 3.65i)T + (2.12 + 60.9i)T^{2} \)
67 \( 1 + (-2.72 - 1.70i)T + (29.3 + 60.2i)T^{2} \)
71 \( 1 + (12.4 + 6.59i)T + (39.7 + 58.8i)T^{2} \)
73 \( 1 + (0.376 - 0.557i)T + (-27.3 - 67.6i)T^{2} \)
79 \( 1 + (-0.311 - 8.93i)T + (-78.8 + 5.51i)T^{2} \)
83 \( 1 + (7.87 - 8.75i)T + (-8.67 - 82.5i)T^{2} \)
89 \( 1 + (-7.72 + 2.21i)T + (75.4 - 47.1i)T^{2} \)
97 \( 1 + (-3.15 + 1.97i)T + (42.5 - 87.1i)T^{2} \)
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

−11.70213031609685640480460849255, −10.45486418913335940021550210247, −9.399042103142779409541430402401, −8.164573836480328688784033008424, −7.14688007927042100177088415300, −6.31263725330726012481662610402, −5.39051049506487401821828379478, −4.32974335509622470209731638248, −3.36541720549460437420952826804, −2.17433440333535150511755721910, 1.36584111635707941128559978947, 3.35391382709232852816606185888, 4.35133447536001981431292023809, 4.75813670311079052203331631729, 5.85268863979284184882426466293, 7.17844121479097846980913640005, 7.962129205868793203972660920704, 9.158764496648787021619350595984, 10.20174060663271623465767721213, 11.19038511953620380010996815907

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