L-function 1-1925-1925.1083-r0-0-0

• 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: $0$

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 + ⋯

Functional equation

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

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

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

Euler product

   $L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$

	$p$	$F_p(T)$
bad	5	$1$
	7	$1$
	11	$1$
good	2	$1 + (0.207 + 0.978i)T$
	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	$1 + (0.406 - 0.913i)T$

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