Properties

Label 2-475-475.6-c1-0-36
Degree $2$
Conductor $475$
Sign $-0.780 - 0.625i$
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  + (−0.695 − 0.434i)2-s + (−2.62 + 0.752i)3-s + (−0.581 − 1.19i)4-s + (−1.90 − 1.17i)5-s + (2.15 + 0.617i)6-s + (2.44 − 4.24i)7-s + (−0.285 + 2.71i)8-s + (3.77 − 2.36i)9-s + (0.815 + 1.64i)10-s + (1.65 − 1.83i)11-s + (2.42 + 2.69i)12-s + (−0.143 − 4.09i)13-s + (−3.54 + 1.88i)14-s + (5.88 + 1.64i)15-s + (−0.255 + 0.327i)16-s + (−6.33 + 0.443i)17-s + ⋯
L(s)  = 1  + (−0.491 − 0.307i)2-s + (−1.51 + 0.434i)3-s + (−0.290 − 0.596i)4-s + (−0.851 − 0.524i)5-s + (0.878 + 0.252i)6-s + (0.925 − 1.60i)7-s + (−0.100 + 0.959i)8-s + (1.25 − 0.787i)9-s + (0.257 + 0.519i)10-s + (0.497 − 0.552i)11-s + (0.699 + 0.777i)12-s + (−0.0396 − 1.13i)13-s + (−0.948 + 0.504i)14-s + (1.51 + 0.424i)15-s + (−0.0639 + 0.0818i)16-s + (−1.53 + 0.107i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.780 - 0.625i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.780 - 0.625i$
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: \((2,\ 475,\ (\ :1/2),\ -0.780 - 0.625i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0852155 + 0.242701i\)
\(L(\frac12)\) \(\approx\) \(0.0852155 + 0.242701i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.90 + 1.17i)T \)
19 \( 1 + (-2.92 + 3.22i)T \)
good2 \( 1 + (0.695 + 0.434i)T + (0.876 + 1.79i)T^{2} \)
3 \( 1 + (2.62 - 0.752i)T + (2.54 - 1.58i)T^{2} \)
7 \( 1 + (-2.44 + 4.24i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.65 + 1.83i)T + (-1.14 - 10.9i)T^{2} \)
13 \( 1 + (0.143 + 4.09i)T + (-12.9 + 0.906i)T^{2} \)
17 \( 1 + (6.33 - 0.443i)T + (16.8 - 2.36i)T^{2} \)
23 \( 1 + (5.68 - 0.798i)T + (22.1 - 6.33i)T^{2} \)
29 \( 1 + (-3.11 - 0.218i)T + (28.7 + 4.03i)T^{2} \)
31 \( 1 + (5.14 - 2.28i)T + (20.7 - 23.0i)T^{2} \)
37 \( 1 + (-0.109 + 0.336i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.63 + 3.37i)T + (-9.91 - 39.7i)T^{2} \)
43 \( 1 + (-1.16 - 6.57i)T + (-40.4 + 14.7i)T^{2} \)
47 \( 1 + (8.52 + 0.595i)T + (46.5 + 6.54i)T^{2} \)
53 \( 1 + (-0.964 - 1.97i)T + (-32.6 + 41.7i)T^{2} \)
59 \( 1 + (-1.52 - 3.76i)T + (-42.4 + 40.9i)T^{2} \)
61 \( 1 + (-7.92 + 1.11i)T + (58.6 - 16.8i)T^{2} \)
67 \( 1 + (-2.27 - 9.13i)T + (-59.1 + 31.4i)T^{2} \)
71 \( 1 + (-7.08 - 6.84i)T + (2.47 + 70.9i)T^{2} \)
73 \( 1 + (-0.277 + 7.94i)T + (-72.8 - 5.09i)T^{2} \)
79 \( 1 + (7.72 - 2.21i)T + (66.9 - 41.8i)T^{2} \)
83 \( 1 + (-5.56 + 2.47i)T + (55.5 - 61.6i)T^{2} \)
89 \( 1 + (2.04 + 2.61i)T + (-21.5 + 86.3i)T^{2} \)
97 \( 1 + (-1.26 + 5.08i)T + (-85.6 - 45.5i)T^{2} \)
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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

−10.67703682548774514048788317006, −9.957589870166750544369734610784, −8.734930824821387063843178258495, −7.81606018883902983639612505032, −6.70515551933050423971024746297, −5.45959573363251111385601839235, −4.71682389176656889003921987943, −4.00168877008161780227862902031, −1.13161908443613197852876218377, −0.27460298965665455836521388754, 2.04693270515289664662800548746, 4.07627303172884069110566102452, 4.96347756198043149213654464990, 6.26124121343392197503261532453, 6.88325795405944965661323850519, 7.87163174258638282330988186823, 8.697655591389224809479005965216, 9.652583788358000537500770666253, 11.08207849672838525066627325654, 11.69048469053853454640794346768

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