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Properties

Label 2-475-5.4-c3-0-14
Degree $2$
Conductor $475$
Sign $-0.447 - 0.894i$
Analytic cond. $28.0259$
Root an. cond. $5.29395$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·2-s − 5i·3-s − 4-s + 15·6-s − 11i·7-s + 21i·8-s + 2·9-s − 54·11-s + 5i·12-s + 11i·13-s + 33·14-s − 71·16-s + 93i·17-s + 6i·18-s − 19·19-s + ⋯
L(s)  = 1  + 1.06i·2-s − 0.962i·3-s − 0.125·4-s + 1.02·6-s − 0.593i·7-s + 0.928i·8-s + 0.0740·9-s − 1.48·11-s + 0.120i·12-s + 0.234i·13-s + 0.629·14-s − 1.10·16-s + 1.32i·17-s + 0.0785i·18-s − 0.229·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(28.0259\)
Root analytic conductor: \(5.29395\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (324, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 475,\ (\ :3/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.567430263\)
\(L(\frac12)\) \(\approx\) \(1.567430263\)
\(L(\frac{5}{2})\) 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 \( 1 + 19T \)
good2 \( 1 - 3iT - 8T^{2} \)
3 \( 1 + 5iT - 27T^{2} \)
7 \( 1 + 11iT - 343T^{2} \)
11 \( 1 + 54T + 1.33e3T^{2} \)
13 \( 1 - 11iT - 2.19e3T^{2} \)
17 \( 1 - 93iT - 4.91e3T^{2} \)
23 \( 1 - 183iT - 1.21e4T^{2} \)
29 \( 1 - 249T + 2.43e4T^{2} \)
31 \( 1 - 56T + 2.97e4T^{2} \)
37 \( 1 - 250iT - 5.06e4T^{2} \)
41 \( 1 - 240T + 6.89e4T^{2} \)
43 \( 1 + 196iT - 7.95e4T^{2} \)
47 \( 1 - 168iT - 1.03e5T^{2} \)
53 \( 1 - 435iT - 1.48e5T^{2} \)
59 \( 1 + 195T + 2.05e5T^{2} \)
61 \( 1 + 358T + 2.26e5T^{2} \)
67 \( 1 - 961iT - 3.00e5T^{2} \)
71 \( 1 + 246T + 3.57e5T^{2} \)
73 \( 1 - 353iT - 3.89e5T^{2} \)
79 \( 1 - 34T + 4.93e5T^{2} \)
83 \( 1 - 234iT - 5.71e5T^{2} \)
89 \( 1 - 168T + 7.04e5T^{2} \)
97 \( 1 + 758iT - 9.12e5T^{2} \)
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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

−10.80314763580801891553656838867, −10.08420992789370395966068649220, −8.548908294762682033419836370368, −7.81864591517462402376587139218, −7.31444853595564664213861105993, −6.41161682205193025384586499593, −5.60910596357799544439445874679, −4.39061502826092428991832276329, −2.65800746886934216575040268246, −1.37157049375896041740276200555, 0.49349360468485387415793609574, 2.38749732175947229116550897311, 3.01816408059376240564816872620, 4.39976894194002726261044658211, 5.13943368524502442667515847757, 6.51055391557236485867852902912, 7.71014582710734509165631205051, 8.896512592007326063975816046177, 9.733329450671153344174543374335, 10.51117018305138200126054120160

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