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Properties

Label 1-475-475.36-r0-0-0
Degree $1$
Conductor $475$
Sign $0.952 - 0.304i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.615 + 0.788i)2-s + (0.438 − 0.898i)3-s + (−0.241 − 0.970i)4-s + (0.438 + 0.898i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.615 − 0.788i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (0.990 − 0.139i)13-s + (−0.374 − 0.927i)14-s + (−0.882 + 0.469i)16-s + (0.961 − 0.275i)17-s + 18-s + (0.559 + 0.829i)21-s + (0.438 − 0.898i)22-s + ⋯
L(s)  = 1  + (−0.615 + 0.788i)2-s + (0.438 − 0.898i)3-s + (−0.241 − 0.970i)4-s + (0.438 + 0.898i)6-s + (−0.5 + 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.615 − 0.788i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (0.990 − 0.139i)13-s + (−0.374 − 0.927i)14-s + (−0.882 + 0.469i)16-s + (0.961 − 0.275i)17-s + 18-s + (0.559 + 0.829i)21-s + (0.438 − 0.898i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.952 - 0.304i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.952 - 0.304i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9960848417 - 0.1551420999i\)
\(L(\frac12)\) \(\approx\) \(0.9960848417 - 0.1551420999i\)
\(L(1)\) \(\approx\) \(0.8627841591 + 0.01043999958i\)
\(L(1)\) \(\approx\) \(0.8627841591 + 0.01043999958i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.615 + 0.788i)T \)
3 \( 1 + (0.438 - 0.898i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (0.990 - 0.139i)T \)
17 \( 1 + (0.961 - 0.275i)T \)
23 \( 1 + (0.848 - 0.529i)T \)
29 \( 1 + (0.961 + 0.275i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.882 + 0.469i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (0.961 + 0.275i)T \)
53 \( 1 + (-0.241 - 0.970i)T \)
59 \( 1 + (0.0348 - 0.999i)T \)
61 \( 1 + (0.848 - 0.529i)T \)
67 \( 1 + (0.559 - 0.829i)T \)
71 \( 1 + (-0.997 + 0.0697i)T \)
73 \( 1 + (0.990 + 0.139i)T \)
79 \( 1 + (0.438 - 0.898i)T \)
83 \( 1 + (-0.104 - 0.994i)T \)
89 \( 1 + (-0.882 - 0.469i)T \)
97 \( 1 + (0.559 + 0.829i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.45816850809859151406015194222, −22.97879724346355306202855953763, −21.7896545384806290137052963452, −21.048562692205705186706070198494, −20.6314384522925658493608291080, −19.60422293670001661122652288368, −19.04893729997215178498065355842, −17.95353641415364779501320470434, −16.92066915015214651963058374043, −16.23813902369646746726071093816, −15.54973519093318705056261016690, −14.06530203712548272832758102968, −13.46514684117546102664485711285, −12.49851563728226248874075958551, −11.13534040100036986501742388945, −10.55941191764836430919916465431, −9.91555708552381217533943664669, −8.92172532281890749871526115110, −8.11212496796486291798829620821, −7.173702338472451962848472218056, −5.54720878821038350846827342137, −4.278974985039107705093419962672, −3.46636074732425223924427951924, −2.70663533696747889990391279377, −1.09480434639311906535780490502, 0.815301084813312091035863445002, 2.168770265474204809871089908362, 3.228669803670302313906784453137, 5.08578754042437455422808553796, 5.99277673345926303572504548305, 6.747456048298918744952384404634, 7.83239787852426530455329969160, 8.45381981488969704474459856230, 9.31531152835040388803216913007, 10.30681022025028043394030189879, 11.532042781211764096831930830727, 12.696860333830680308324069149858, 13.38921188278629529988552608073, 14.37187282234806844492005025000, 15.279764037286300114419276081678, 15.93345568241534428077757141462, 16.99653470242144208150505328665, 18.02580729096452399480983717782, 18.73992505393989988585093837387, 18.93994783098330263413766695352, 20.2303993835120851866828108768, 20.95265709880935385844454154750, 22.43959712546526387957985113633, 23.32068603787060521928141232984, 23.81482100563302397813377120805

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