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Properties

Label 2-475-95.8-c1-0-4
Degree $2$
Conductor $475$
Sign $-0.970 + 0.240i$
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  + (0.633 + 2.36i)3-s + (−1.73 + i)4-s + (−2 + 2i)7-s + (−2.59 + 1.50i)9-s − 11-s + (−3.46 − 3.46i)12-s + (−2.36 − 0.633i)13-s + (1.99 − 3.46i)16-s + (−1.09 − 4.09i)17-s + (4.33 − 0.5i)19-s + (−6 − 3.46i)21-s + (−1.46 + 5.46i)23-s + (1.46 − 5.46i)28-s + (−0.866 − 1.5i)29-s + 8.66i·31-s + ⋯
L(s)  = 1  + (0.366 + 1.36i)3-s + (−0.866 + 0.5i)4-s + (−0.755 + 0.755i)7-s + (−0.866 + 0.500i)9-s − 0.301·11-s + (−0.999 − 0.999i)12-s + (−0.656 − 0.175i)13-s + (0.499 − 0.866i)16-s + (−0.266 − 0.993i)17-s + (0.993 − 0.114i)19-s + (−1.30 − 0.755i)21-s + (−0.305 + 1.13i)23-s + (0.276 − 1.03i)28-s + (−0.160 − 0.278i)29-s + 1.55i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 + 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.970 + 0.240i$
Analytic conductor: \(3.79289\)
Root analytic conductor: \(1.94753\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :1/2),\ -0.970 + 0.240i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0891223 - 0.730423i\)
\(L(\frac12)\) \(\approx\) \(0.0891223 - 0.730423i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 + (-4.33 + 0.5i)T \)
good2 \( 1 + (1.73 - i)T^{2} \)
3 \( 1 + (-0.633 - 2.36i)T + (-2.59 + 1.5i)T^{2} \)
7 \( 1 + (2 - 2i)T - 7iT^{2} \)
11 \( 1 + T + 11T^{2} \)
13 \( 1 + (2.36 + 0.633i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (1.09 + 4.09i)T + (-14.7 + 8.5i)T^{2} \)
23 \( 1 + (1.46 - 5.46i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (0.866 + 1.5i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.66iT - 31T^{2} \)
37 \( 1 + (-3.46 - 3.46i)T + 37iT^{2} \)
41 \( 1 + (9 + 5.19i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (10.9 - 2.92i)T + (37.2 - 21.5i)T^{2} \)
47 \( 1 + (6.83 + 1.83i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.36 - 0.633i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (0.866 - 1.5i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.5 - 6.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.633 + 2.36i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-7.5 - 4.33i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-2.73 + 0.732i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.59 - 4.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83iT^{2} \)
89 \( 1 + (-6.06 - 10.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.46 - 2.53i)T + (84.0 - 48.5i)T^{2} \)
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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

−11.54473676706964453656355374718, −10.09273626010232836040244356090, −9.691657536988562372461563633722, −9.061798364428990082654800104939, −8.211023648088661629712485425069, −7.02030675873192717083941645936, −5.32250508128364480329283485801, −4.89335176815721131915763373119, −3.53443123454865740533719985096, −2.94367650037530551376972800908, 0.43141828203135188143038543647, 1.91657498685519080018482092223, 3.47934768702709449299094979933, 4.74486587098875870812952531646, 6.08056953610029820667500758631, 6.83494534823652798190123119286, 7.82079323609303669575528495344, 8.533796675976933018077933190842, 9.743750196454801994008859321497, 10.24140581079916300502676285266

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