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Properties

Label 2-475-475.6-c1-0-9
Degree $2$
Conductor $475$
Sign $0.973 - 0.227i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.918 − 0.574i)2-s + (−1.08 + 0.311i)3-s + (−0.362 − 0.742i)4-s + (−0.569 − 2.16i)5-s + (1.17 + 0.337i)6-s + (−1.35 + 2.34i)7-s + (−0.320 + 3.04i)8-s + (−1.46 + 0.913i)9-s + (−0.717 + 2.31i)10-s + (0.234 − 0.260i)11-s + (0.625 + 0.694i)12-s + (0.0306 + 0.877i)13-s + (2.59 − 1.37i)14-s + (1.29 + 2.17i)15-s + (1.02 − 1.31i)16-s + (5.64 − 0.394i)17-s + ⋯
L(s)  = 1  + (−0.649 − 0.405i)2-s + (−0.627 + 0.179i)3-s + (−0.181 − 0.371i)4-s + (−0.254 − 0.966i)5-s + (0.480 + 0.137i)6-s + (−0.512 + 0.887i)7-s + (−0.113 + 1.07i)8-s + (−0.487 + 0.304i)9-s + (−0.226 + 0.731i)10-s + (0.0705 − 0.0784i)11-s + (0.180 + 0.200i)12-s + (0.00850 + 0.243i)13-s + (0.693 − 0.368i)14-s + (0.333 + 0.560i)15-s + (0.256 − 0.327i)16-s + (1.36 − 0.0957i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.227i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.973 - 0.227i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.973 - 0.227i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.522210 + 0.0602057i\)
\(L(\frac12)\) \(\approx\) \(0.522210 + 0.0602057i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.569 + 2.16i)T \)
19 \( 1 + (-4.32 + 0.522i)T \)
good2 \( 1 + (0.918 + 0.574i)T + (0.876 + 1.79i)T^{2} \)
3 \( 1 + (1.08 - 0.311i)T + (2.54 - 1.58i)T^{2} \)
7 \( 1 + (1.35 - 2.34i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-0.234 + 0.260i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (-0.0306 - 0.877i)T + (-12.9 + 0.906i)T^{2} \)
17 \( 1 + (-5.64 + 0.394i)T + (16.8 - 2.36i)T^{2} \)
23 \( 1 + (4.73 - 0.665i)T + (22.1 - 6.33i)T^{2} \)
29 \( 1 + (3.35 + 0.234i)T + (28.7 + 4.03i)T^{2} \)
31 \( 1 + (-8.11 + 3.61i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (2.08 - 6.40i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.28 + 4.20i)T + (-9.91 - 39.7i)T^{2} \)
43 \( 1 + (-2.07 - 11.7i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (-13.0 - 0.910i)T + (46.5 + 6.54i)T^{2} \)
53 \( 1 + (-1.97 - 4.05i)T + (-32.6 + 41.7i)T^{2} \)
59 \( 1 + (-0.641 - 1.58i)T + (-42.4 + 40.9i)T^{2} \)
61 \( 1 + (15.2 - 2.14i)T + (58.6 - 16.8i)T^{2} \)
67 \( 1 + (-1.09 - 4.38i)T + (-59.1 + 31.4i)T^{2} \)
71 \( 1 + (-9.87 - 9.53i)T + (2.47 + 70.9i)T^{2} \)
73 \( 1 + (0.239 - 6.86i)T + (-72.8 - 5.09i)T^{2} \)
79 \( 1 + (4.86 - 1.39i)T + (66.9 - 41.8i)T^{2} \)
83 \( 1 + (-0.0608 + 0.0271i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (2.32 + 2.97i)T + (-21.5 + 86.3i)T^{2} \)
97 \( 1 + (-0.348 + 1.39i)T + (-85.6 - 45.5i)T^{2} \)
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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

−11.07300471071838410945264062813, −9.916723706321925189682942869648, −9.463931086835010660088556538512, −8.532467295453604217770249484322, −7.78081123347314950396654549503, −5.82408334986699504435923880769, −5.64514914673858353252666790889, −4.47715821679451811072222564900, −2.71868330438929845865770941320, −1.04924361088086751696593312328, 0.55923079679323111258179262330, 3.17022223907018683621358437669, 3.89099101679226147985268988296, 5.66495414518566504840317953241, 6.58413279228625062725436345786, 7.37233703226543895626207074727, 7.961871617589627968553508409661, 9.248452227812687394123932964261, 10.14524631369223087714573489414, 10.71787368634262753223567855522

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