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Properties

Label 1-1925-1925.1083-r0-0-0
Degree $1$
Conductor $1925$
Sign $-0.993 + 0.109i$
Analytic cond. $8.93966$
Root an. cond. $8.93966$
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  + (0.207 + 0.978i)2-s + (0.866 + 0.5i)3-s + (−0.913 + 0.406i)4-s + (−0.309 + 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.5 + 0.866i)9-s + (−0.994 − 0.104i)12-s − i·13-s + (0.669 − 0.743i)16-s + (−0.743 + 0.669i)17-s + (−0.743 + 0.669i)18-s + (0.913 + 0.406i)19-s + (−0.406 + 0.913i)23-s + (−0.104 − 0.994i)24-s + (0.978 − 0.207i)26-s + i·27-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (0.866 + 0.5i)3-s + (−0.913 + 0.406i)4-s + (−0.309 + 0.951i)6-s + (−0.587 − 0.809i)8-s + (0.5 + 0.866i)9-s + (−0.994 − 0.104i)12-s − i·13-s + (0.669 − 0.743i)16-s + (−0.743 + 0.669i)17-s + (−0.743 + 0.669i)18-s + (0.913 + 0.406i)19-s + (−0.406 + 0.913i)23-s + (−0.104 − 0.994i)24-s + (0.978 − 0.207i)26-s + i·27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1925 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.109i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1925\)    =    \(5^{2} \cdot 7 \cdot 11\)
Sign: $-0.993 + 0.109i$
Analytic conductor: \(8.93966\)
Root analytic conductor: \(8.93966\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1925} (1083, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1925,\ (0:\ ),\ -0.993 + 0.109i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1037520597 + 1.890039431i\)
\(L(\frac12)\) \(\approx\) \(0.1037520597 + 1.890039431i\)
\(L(1)\) \(\approx\) \(0.9280558161 + 0.9814432559i\)
\(L(1)\) \(\approx\) \(0.9280558161 + 0.9814432559i\)

Euler product

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

−19.81117088594578071286660867278, −19.007591172507007541453027752401, −18.48064023458066101021066435216, −17.90538138465367706995371886992, −16.95170718719834941864364842343, −15.81725856193477409604582633881, −15.08125331498284240799713432667, −14.106159319659876080600227697149, −13.81564088314190298774845466468, −13.13801225037335537663548152492, −12.137165322567058088661327592994, −11.79192647056365249435568231139, −10.78781267295350818231243030367, −9.88901474982765394237202177170, −9.11940720458665680133757531680, −8.73699908752930677789194570600, −7.66445091866527892283490465034, −6.8338708868277549550322368353, −5.906122252497717411106615270663, −4.66917332561579919176474758260, −4.11571273928624678281108362967, −3.09455403444286389685497245104, −2.40310778676768066411627765130, −1.67363397858922194616241185704, −0.55772155800217477509235544260, 1.32096179853214973888828453245, 2.69084543166953081312875687261, 3.51481609280689224858907689737, 4.15653379816293862993172246724, 5.16472595892635755888780699553, 5.73737588459429945264485516999, 6.8879452884915813763995534785, 7.642526975599482598600494631391, 8.27312238458212133153262573641, 8.9398337414318834914257589418, 9.77781512122366458906936742030, 10.37915253337800359763553848200, 11.49615264011998052843797487018, 12.74057773850252096180368859220, 13.12766082786597023537692007383, 14.01168095850245554333789250979, 14.610970545230049343979136877164, 15.2336861156393061746823287319, 15.99020362457629866442922541736, 16.34084928148432715563778899681, 17.542613323160265028125398284867, 17.9674036685606677597996334900, 18.875729257193755223377895053097, 19.843235937617636994991380714215, 20.22487423621575382465540830306

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