Properties

Label 2-475-475.396-c1-0-7
Degree $2$
Conductor $475$
Sign $0.643 - 0.765i$
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  + (−2.19 + 1.36i)2-s + (−1.37 − 0.395i)3-s + (2.05 − 4.20i)4-s + (−2.03 + 0.933i)5-s + (3.56 − 1.02i)6-s + (1.36 + 2.36i)7-s + (0.724 + 6.89i)8-s + (−0.799 − 0.499i)9-s + (3.17 − 4.82i)10-s + (−2.40 − 2.66i)11-s + (−4.49 + 4.98i)12-s + (0.0645 − 1.84i)13-s + (−6.23 − 3.31i)14-s + (3.17 − 0.483i)15-s + (−5.26 − 6.73i)16-s + (0.321 + 0.0225i)17-s + ⋯
L(s)  = 1  + (−1.54 + 0.968i)2-s + (−0.795 − 0.228i)3-s + (1.02 − 2.10i)4-s + (−0.908 + 0.417i)5-s + (1.45 − 0.417i)6-s + (0.516 + 0.893i)7-s + (0.256 + 2.43i)8-s + (−0.266 − 0.166i)9-s + (1.00 − 1.52i)10-s + (−0.724 − 0.804i)11-s + (−1.29 + 1.44i)12-s + (0.0178 − 0.512i)13-s + (−1.66 − 0.885i)14-s + (0.818 − 0.124i)15-s + (−1.31 − 1.68i)16-s + (0.0780 + 0.00545i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.643 - 0.765i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (396, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ 0.643 - 0.765i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.290660 + 0.135467i\)
\(L(\frac12)\) \(\approx\) \(0.290660 + 0.135467i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.03 - 0.933i)T \)
19 \( 1 + (4.02 + 1.67i)T \)
good2 \( 1 + (2.19 - 1.36i)T + (0.876 - 1.79i)T^{2} \)
3 \( 1 + (1.37 + 0.395i)T + (2.54 + 1.58i)T^{2} \)
7 \( 1 + (-1.36 - 2.36i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (2.40 + 2.66i)T + (-1.14 + 10.9i)T^{2} \)
13 \( 1 + (-0.0645 + 1.84i)T + (-12.9 - 0.906i)T^{2} \)
17 \( 1 + (-0.321 - 0.0225i)T + (16.8 + 2.36i)T^{2} \)
23 \( 1 + (2.55 + 0.359i)T + (22.1 + 6.33i)T^{2} \)
29 \( 1 + (-3.37 + 0.236i)T + (28.7 - 4.03i)T^{2} \)
31 \( 1 + (-8.51 - 3.79i)T + (20.7 + 23.0i)T^{2} \)
37 \( 1 + (-0.407 - 1.25i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-5.22 - 6.68i)T + (-9.91 + 39.7i)T^{2} \)
43 \( 1 + (-0.627 + 3.55i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-3.45 + 0.241i)T + (46.5 - 6.54i)T^{2} \)
53 \( 1 + (4.61 - 9.46i)T + (-32.6 - 41.7i)T^{2} \)
59 \( 1 + (0.613 - 1.51i)T + (-42.4 - 40.9i)T^{2} \)
61 \( 1 + (-0.426 - 0.0599i)T + (58.6 + 16.8i)T^{2} \)
67 \( 1 + (3.09 - 12.4i)T + (-59.1 - 31.4i)T^{2} \)
71 \( 1 + (-9.57 + 9.24i)T + (2.47 - 70.9i)T^{2} \)
73 \( 1 + (0.346 + 9.91i)T + (-72.8 + 5.09i)T^{2} \)
79 \( 1 + (-15.4 - 4.41i)T + (66.9 + 41.8i)T^{2} \)
83 \( 1 + (-11.7 - 5.24i)T + (55.5 + 61.6i)T^{2} \)
89 \( 1 + (-7.56 + 9.68i)T + (-21.5 - 86.3i)T^{2} \)
97 \( 1 + (0.505 + 2.02i)T + (-85.6 + 45.5i)T^{2} \)
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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

−10.86944845915264397553340507657, −10.37721992594859812068578356751, −8.945180832830304942741658203383, −8.311620522895032302800851226937, −7.75413583972271887324022739963, −6.52634793950187140493946804588, −5.99970078039444769239081814459, −4.95792406064989732045284625885, −2.71925072139540503306579964693, −0.65217430570256414949885130947, 0.67359905835992660239934528359, 2.27380377627392398275291957518, 3.94760533011111876025453523457, 4.82665416147442319501458047906, 6.62545565626436230884057217239, 7.88100673556581498075443933153, 8.023827983696801339568089481096, 9.245621292187969480476455032176, 10.33070766017681235040219256773, 10.70779897240321324071129446637

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