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Properties

Label 1-475-475.103-r0-0-0
Degree $1$
Conductor $475$
Sign $0.374 + 0.927i$
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  + (−0.994 − 0.104i)2-s + (−0.743 − 0.669i)3-s + (0.978 + 0.207i)4-s + (0.669 + 0.743i)6-s − i·7-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.809 − 0.587i)11-s + (−0.587 − 0.809i)12-s + (−0.994 + 0.104i)13-s + (−0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (0.207 + 0.978i)17-s − i·18-s + (−0.669 + 0.743i)21-s + (0.743 + 0.669i)22-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.743 − 0.669i)3-s + (0.978 + 0.207i)4-s + (0.669 + 0.743i)6-s − i·7-s + (−0.951 − 0.309i)8-s + (0.104 + 0.994i)9-s + (−0.809 − 0.587i)11-s + (−0.587 − 0.809i)12-s + (−0.994 + 0.104i)13-s + (−0.104 + 0.994i)14-s + (0.913 + 0.406i)16-s + (0.207 + 0.978i)17-s − i·18-s + (−0.669 + 0.743i)21-s + (0.743 + 0.669i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.374 + 0.927i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.374 + 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1972096554 + 0.1330416690i\)
\(L(\frac12)\) \(\approx\) \(0.1972096554 + 0.1330416690i\)
\(L(1)\) \(\approx\) \(0.4238880647 - 0.09336841704i\)
\(L(1)\) \(\approx\) \(0.4238880647 - 0.09336841704i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.994 - 0.104i)T \)
3 \( 1 + (-0.743 - 0.669i)T \)
7 \( 1 - iT \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.994 + 0.104i)T \)
17 \( 1 + (0.207 + 0.978i)T \)
23 \( 1 + (0.406 + 0.913i)T \)
29 \( 1 + (-0.978 - 0.207i)T \)
31 \( 1 + (-0.309 + 0.951i)T \)
37 \( 1 + (-0.587 - 0.809i)T \)
41 \( 1 + (-0.913 - 0.406i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 + (-0.207 + 0.978i)T \)
53 \( 1 + (-0.207 + 0.978i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.743 + 0.669i)T \)
71 \( 1 + (-0.669 + 0.743i)T \)
73 \( 1 + (-0.994 - 0.104i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.951 + 0.309i)T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.743 + 0.669i)T \)
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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

−23.83628150960223751748975026554, −22.651405937690844296427869409351, −21.99906652884532624451143762673, −20.83349337724521964198585409225, −20.51392461290844439657664702190, −19.12445853136661630451228362764, −18.33996790858679689564320711225, −17.75502751641502099833009610168, −16.756749863275307849832201801223, −16.12634946552306864896025124613, −15.13937371530068615627906185357, −14.80464062258297259766150152829, −12.815433477656436501067350553254, −11.9763299126706350920685483920, −11.306571769024042630787779317675, −10.192296868693673477864138518335, −9.646555017354207480789968728035, −8.76369534114846372885625953708, −7.57735661904482191312736619699, −6.63870710149253432437805478601, −5.49397894605918081880753300292, −4.879684088200219574797591095524, −3.07005491437310031407111445951, −2.08466490312456381544866156276, −0.220190631872656622280784965086, 1.1064938843578460996838863318, 2.19476512029193499976584743304, 3.570343238791978512702328878287, 5.179867908804195010418443796035, 6.15073346716141888726080096069, 7.36504036886395240872162672, 7.56776122089794778158033689026, 8.82389933240867476737903737400, 10.17227817078773960072187197937, 10.67068308920109867236654094606, 11.5134017305362419062513211537, 12.52861078697255195131525702295, 13.28490898361322879455299425001, 14.487777084530894359416441933612, 15.77742328934437594799217899923, 16.579371110632401462825383691641, 17.320412438392162625340473087040, 17.76436529691694182055363340679, 19.10259886025871132191749262389, 19.23981816770410010263986295912, 20.38878028666499892638862526836, 21.35727350899292342235259281165, 22.227049663724864156496991883898, 23.573046579674000637302381117516, 23.879310688977362633226033463243

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